Let us first consider the practice of our own churches.
If we have a children’s ministry, teach and catechize our children, periodically call our children disciples, and let our children worship in “big church,” then, yes, we should baptize our children.
If this is our practice, then it’s clear that we don’t exclude our children from participation in the life of the church; we include them. In this case our children belong to the church. And if it’s true that our children are included in/belong to the church, then it follows that our children should receive the sign of entrance into the church: baptism. If in practice we’re already counting them as members of the church, why wouldn’t we give them the sign of entrance into the church?
Let us now consider the issue in light of the Bible’s teaching about the Abrahamic covenant.
In Peter’s Pentecost sermon, he tells people to repent and be baptized because “the promise is for you and for your children and for all who are far off” (Acts 2:39). “The promise” refers to the baptism of the Holy Spirit in Acts 1:4-5 and applies to those who repent/believe, their children, and everyone whom God calls to Himself. (This borrows language from the Abrahamic covenant in Genesis 12, 15, and 17, which applies to Abraham, his children, and the foreigners who enter his house.)
Then Paul, in Galatians 3:13-14, says that the promise of the Holy Spirit is the blessing promised to Abraham. Paul concludes that “if you are Christ’s, then you are Abraham’s offspring, heirs according to the promise” (Galatians 3:29). “The promise” here refers to the covenant God made with Abraham in Genesis 12, 15, and 17, which Paul says applies to new testament Christians. In Genesis 17, God establishes His covenant with Abraham, Abraham’s children (v. 7), and the foreigners who come into Abraham’s house (v. 12).
So, following Paul’s argument in Galatians 3, new testament Christians belong to the same covenant to which Abraham belonged. This means that our children also belong to this covenant since the covenant promises made to Abraham applied also to his children; the signs and promises applied to Isaac just as much as they did to Abraham. The promises given to Abraham (and his children) are now ours (and our children’s) in Christ.
The importance of all this for baptism is that the original sign of entrance into the Abrahamic covenant (circumcision) applied both to Abraham after he believed (cf. Romans 4:10-11) and to Abraham’s children before they believed (cf. Genesis 17:12). So the signs of the Abrahamic covenant apply to believers and their children, but in slightly different ways. Following the biblical pattern, adult converts who have yet to receive the sign of entrance into the covenant community should receive the sign after they believe, but the children of believers should receive the sign of entrance into the covenant community before they believe. And, again, if we are part of the Abrahamic covenant, then the signs apply to us and to our children as well.
If we’re converted in adulthood (as I was), then we should receive the sign of entrance into the covenant community after we believe (as I did); but since the covenant promises apply equally to our children, our children should, following the Abrahamic pattern, receive the sign of entrance into the covenant community before they believe. In biblical terms, my children will belong to the covenant community by virtue of the fact that God’s promises in Christ are for me and my children. They’re counted as members of the covenant community because they belong to me, a member of the covenant community. That’s the pattern of the Abrahamic covenant, to which I (and all other believers) now belong. Thus, since my children are counted as members of the covenant community, they should receive the sign of entrance into it.
In Colossians 2:11-12, Paul tells us that where circumcision was the original sign of entrance into the covenant community, now baptism is the new sign of entrance into the covenant community. Thus just as Abraham circumcised his children so that they became part of the visible covenant community (Israel), so too we should baptize our children so that they become part of the visible covenant community (the church).
So wrapping everything up: Christians and our children are members of the Abrahamic covenant. The promises and signs given to Abraham and his children thus apply to us and to our children. Since baptism is the new sign of entrance into the Abrahamic covenant, it therefore ought to be administered to us and to our children.