The Better Promises of the New Covenant to the Children of Believers
Inclusion on the Basis of the Believing Parent vs. Exclusion on the Basis of the Unbelieving Parent
By definition, the new covenant is enacted on better promises than the old covenant (Heb. 8:6). But does this mean that all God's promises in the new covenant better than God's promises in the old covenant? Does God weaken or turn away from any of his promises, now that Christ has come into the world?
Specifically, what about God's promises for the children of believers? In the old covenant, God promised to be God not only to his people, but to their children as well (Gen. 17:7). Do the children of believers now have fewer, inferior promises under the new covenant than they did under the covenant?
Absolutely not. The Scriptures are clear that God's promises to the children of believers are better under the new covenant than under the old.
The Old Covenant: Good Promises for the Children of Believers
As stated earlier, God promised explicitly to be God to the children his people:
And I will establish my covenant between me and you and your offspring after you throughout their generations for an everlasting covenant, to be God to you and to your offspring after you. (Gen. 17:7)
Certainly, this is a wonderful promise. Still, this promise had serious limitations under the old covenant.
Deuteronomy 23: Children of Mixed Marriages Excluded from the Old Covenant Church
When there were mixed marriages with foreigners, the children of those foreigners could not be included in the assembly of the Lord for several generations. Some children could never enter the assembly of the Lord, even to the tenth generation:
 “No one born of a forbidden union may enter the assembly of the LORD. Even to the tenth generation, none of his descendants may enter the assembly of the LORD.  “No Ammonite or Moabite may enter the assembly of the LORD. Even to the tenth generation, none of them may enter the assembly of the LORD forever,  because they did not meet you with bread and with water on the way, when you came out of Egypt, and because they hired against you Balaam the son of Beor from Pethor of Mesopotamia, to curse you.” (Deut. 23:2–4)
Other children of believers were permitted to enter the assembly of the Lord, but only after three generations:
 “You shall not abhor an Edomite, for he is your brother. You shall not abhor an Egyptian, because you were a sojourner in his land.  Children born to them in the third generation may enter the assembly of the LORD. (Deut. 23:7–8)
In these passages, the Hebrew word for assembly is qāhal. In the Greek translation of the Old Testament, this word is rendered as ekklēsia—the exact same word that the New Testament uses for “church.”
Thus, the children born of mixed marriages between God's people and foreigners were not permitted to be members of the old covenant church for several generations—or, sometimes, forever.
Malachi 2: Children of Foreign Marriages Cut off from Jacob's Tents
The same idea comes up in Malachi 2, where God rebukes Judah for marrying the daughter of a foreign god (Mal. 2:11). In response, God excommunicates ("cuts off") the children of these marriages from the membership of his people:
 Judah has been faithless, and abomination has been committed in Israel and in Jerusalem. For Judah has profaned the sanctuary of the LORD, which he loves, and has married the daughter of a foreign god.  May the LORD cut off from the tents of Jacob any descendant of the man who does this, who brings an offering to the LORD of hosts! (Mal. 2:11–12)
The "tents of Jacob" is a poetic way of describing the nation of Israel—that is, the old covenant assembly (church) of God. Here again, God explicitly cuts off the children of mixed marriages. This passage does not introduce a new idea, but simply applies the law from Deuteronomy to the Israelites who were sinning by their intermarriages.
Here, though, God makes his reasons more clear: these are not simply foreigners, but they are the children of other gods. That is, they are worshipers of other gods who will draw God's people away from the living God.
The Counter-Examples: Rahab and Ruth Engrafted by Faith
By observing the God is cutting off the the descendants of those who marry the daughters of foreign gods, we can better understand the major counter-examples of the old covenant: Rahab and Ruth. Both of these women explicitly renounce their foreign gods and put their faith in Yahweh, the God of Israel (Josh. 2:8–13; Ruth 1:16–17).
On the basis of their faith, they are immediately grafted into the covenant people of God (cf. Josh. 6:25; Ruth 2:10–12). Furthermore, their offspring are immediately included in the old covenant church of God. This is a critical point, since we learn that both Rahab and Ruth become mothers in the genealogy that leads to David (Ruth 4:13–22) and ultimately even to Jesus (Matt. 1:5).
God's point is to insist that his people marry within the covenant, for God is seeking "godly offspring" (Mal. 2:15) from their marriages. The marriages whose descendants are cut off from the old covenant church are only those marriages where the wife remains a daughter of a foreign god, refusing to put her faith in Yahweh.
Such marriages would turn Yahweh's people away from their God, to serve foreign gods (Deut. 7:3–4). Whether the daughter of a foreign god was a slave (Gen. 16:3), or was taken in battle (Deut. 21:10–14), or a political ally (1 Kgs. 11:1–8), the children of those marriages were cut off from the promises of the old covenant and excluded from the old covenant church of God.
The New Covenant: Better Promises for the Children of Believers
What about the new covenant? What kind of promises to the children of believers have now that Christ has come into the world? While some believe that God now excludes the children of believers from the new covenant until those children are able to profess faith in Christ, that isn't what the Bible teaches.
Acts 2:39: The Promise is for Your Children
Right away, in the first sermon preached at Pentecost, the Apostle Peter declares that the promises of the gospel are for the children of believers:
 And Peter said to them, “Repent and be baptized every one of you in the name of Jesus Christ for the forgiveness of your sins, and you will receive the gift of the Holy Spirit.  For the promise is for you and for your children and for all who are far off, everyone whom the Lord our God calls to himself.” (Acts 2:38–39)
The promises of the gospel are not made to all indiscriminately, regardless of whose children they were. Instead, Peter specifically insists that the promises are, in a special way, "for" the children of believers. In the context of the whole Bible, the message is clear: the God who made old covenantal promises with the children of believers is still making promises to the children of new covenant believers.
As a confirmation of this principle, we then see the New Testament addressing children as members of their respective churches (Eph. 6:1–3; Col. 3:20; 1 Pet. 5:5). The children of believers are counted as members of God's church under the new covenant, just as they were under the old covenant.
1 Corinthians 7:14: The Children of Mixed Marriages are Holy
But what about the children of mixed marriages under the new covenant?
In 1 Corinthians 7:14, we see how the promises to the children of believers are better under the new covenant than they were under the old covenant. There, the Apostle Paul is addressing whether formerly-pagan, Gentile converts to Christianity should divorce their still-pagan spouses. These are not Rahabs or Ruths—these are the unclean, persistently unbelieving spouses whose children God excluded from his old covenant church.
Surprisingly, Paul urges believing spouses to remain with their unbelieving spouses, if possible. Curiously, the rationale has to do with their children:
For the unbelieving husband is made holy because of his wife, and the unbelieving wife is made holy because of her husband. Otherwise your children would be unclean, but as it is, they are holy. (1 Cor. 7:14)
Somehow, the unbelieving spouse is not counted as defiling the children, as in the old covenant. Instead. the unbelieving spouse is "made holy" because of the believing spouse. While under old covenant logic, the children would be counted as unclean, they are counted as holy under new covenant logic.
Against the backdrop of the old covenant, this is shocking. The children of mixed marriages will be counted holy, rather than being excluded from the assembly of Yahweh and the tents of Jacob? These children are the full recipients of God's promises to his people.
Here is how the Book of Church Order of the Presbyterian Church in America applies this principle:
The children of believers are, through the covenant and by right of birth, non-communing members of the church. Hence they are entitled to Baptism, and to the pastoral oversight, instruction and government of the church, with a view to their embracing Christ and thus possessing personally all benefits of the covenant. (BCO 6-1)
Whether one or both parents are believers, God includes these children as members of the new covenant church. Indeed, the new covenant is enacted upon better promises for the children of believers than the old covenant was.
Postscript: Unbelieving Spouses
What, though, about the unbelieving spouses? Paul warns us not to become unequally yoked by marrying an unbeliever (2 Cor. 6:14). He asks, hauntingly, "For how do you know, wife, whether you will save your husband? Or how do you know, husband, whether you will save your wife?" (1 Cor. 7:16). Echoing the Old Testament, the New Testament still urges believers not to marry unbelievers, even if the children of new covenant believers have better promises than the children of old covenant believers.
What exactly does Paul mean, though, when he tells us that unbelieving spouses are "made holy" because of the believing spouse? Does this mean that the unbelieving spouses are made members of the church too?
T. E. Peck offers this helpful explanation of this difficult question:
The very words teach that this sanctification regards the unbelieving parent, not for his own sake, but as a medium, affecting the transmission of covenant privileges to the children of a believer. The question was, whether, in the case of one of the parties in the marriage-relation being a Pagan, and the other a Christian, the former or the latter should determine the relation of the offspring to the church, or whether neither should. The answer is, that in this case, where the argument for the children seems to be perfectly balanced by the argument against them, God has graciously inclined the scale in favor of his people; so that, for the purpose of conveying to their infants the privilege of being within his covenant and church, the unbelieving partner is sanctified by the believing. It must be thus or the reverse. (Peck, Notes on Ecclesiology, 48–49).
The question still comes back to the application of God's covenant promises to the children of believers. The unbelieving spouse is made holy not for his/her own sake, but for the sake of the believing spouse's children.