Here's a slippery wicket: does God have grandchildren?
Most orthodox Christians believe that there must be a regenerative work of the Holy Spirit in the life of a person before they are finally admitted into the family of God. This work is described in various ways all throughout the New Testament but refers to that moment of moving from unbelief to belief. This work is compelled by, and accomplished by, the Holy Spirit working in our hearts. When that regenerative work occurs, we move from being an old creation to a new creation, or from death to life, or from darkness to light, or any number of other metaphors that are used in the Bible to describe the difference between our former state and our new state. Again, the regeneration is necessary, and it is a work of the Holy Spirit.
This is important: you don't get into the kingdom or family of God just because you are born into a Christian family with Christian parents. There must be a personal belief–putting ones confidence in the work of Jesus Christ for their own salvation–likely followed by some form of profession.
If memory serves me correctly, it was Billy Graham who either coined or popularized the phrase that "God does not have grandchildren." His point was to emphasize the importance of making a personal profession of faith in Jesus Christ; it was this "personal profession" that would indicate that a person possessed saving faith. In that sense, he is correct: God does not have grandchildren. He simply has children, all of whom have professed faith in Jesus Christ.
But what about the children of believing parents? Here we are left with three options.
The first option is that they, too, are Christians, or children of God. There are a multitude of problems with this, beginning with the aforementioned need for regeneration, and ending with the plethora of kids who have unfortunately been born to Christian parents who have then lived a life that ultimately professed a radically different belief. It is clear that simply being born into a Christian family is not enough to save; we must still admit our need for a savior and put our own confidence in Jesus Christ.
The second option is that these children are not Christians at all, but totally pagan, unregenerate, unbelievers, and far from God. We'll deal with this in a moment.
The third option is that these children are in some sort of a different category altogether. Perhaps not "regenerated" and thus fully saved, but not quite pagan, either.
Most Christians fall into the third category, at least in practice. Our children are raised to call Jesus, "Lord". They are raised to call God, "Father". They are taught to follow Jesus, to share Jesus with their friends, to live for Jesus, to pray to Jesus, and to abide by God's word. Of course, we wouldn't require (or should not require) any of these things for a person who was legitimately an unbelieving person. We may want them to live according to God's ideals, we may appeal to them on behalf of Christ, but we will only expect them to actually live that way once they have put their faith in Christ, have been transformed by the Holy Spirit, and genuine love for God and love for their neighbor is actually possible. To force a non-believing person to "act like a Christian" would be to create some sort of moralist, ethical behavior, that is not in any way based on the Gospel. We might create nice people, but not Christians.
But, of course, most of us don't view our children as being exactly the same as unbelievers. We recognize that by virtue of being raised to Christian parents who love Jesus, and teach Jesus in their homes, and call God "Father" themselves, our children receive some sort of benefit of being amongst the people of God. They are treated like they are a "part of the family", even if they haven't been able to profess Jesus Christ on their own. We look forward to the day when they will make that profession, but until then, our confidence is placed fully in Jesus that he will save our children, so we teach them to love Jesus as their savior so that they are prepared to hear his voice when he calls.
So what does that have to do with baptism?
We believe that baptism is a sign of the new covenant. This New Covenant is instituted through Christ's shed blood on the cross, thus making a way for all people to access the family of God through faith in Jesus Christ and confidence in his work on their behalf. Thus, baptism is ultimately a symbol of association with Jesus Christ in his life, death, and resurrection; this is the association that makes the covenant community the covenant community. It is the reason that when we practice adult baptisms, and an adult "receives the sign of the covenant", that we practice baptism by immersion into water. The imagery of the descent into the water and the reemergence from it reminds us that we have been buried with Christ and will rise with Christ–again, through association with his life, death, and resurrection, in faith.
The promises don't stop there, however. We also believe that when Peter told the Jewish people (and visiting foreigners in Jerusalem) that the promises of the Gospel were for them and their children, that he meant it exactly the way they would have taken it. If they, in faith, believed in the promises of God, then their children would receive the benefit of being raised in that covenant family as well. And, additionally, that if their children believed, then they, too, would ultimately receive salvation. In the meantime, the children would be raised to be "part of the family".
It is precisely because these children are considered to be "part of the family" that we baptize our infants, pointing towards Christ as the ultimate means of salvation, yet fully living in the grace of God that our children will be raised as a part of his covenant family. Our children, too, can call God, "Father". They will be taught to abide by His word and to love Jesus as their savior. In the meantime, we continually pray that Jesus would do the difficult work of transforming their hearts by sending the Holy Spirit so that they become a "new creation", through faith in Jesus Christ. When that happens, then we celebrate with them at their profession; what we are celebrating is not something that they have done, but something that God has done, and something we always believed he would.
Furthermore, we see these great promises of God towards his people all throughout Scripture. As we baptize our children because they will be raised as "part of the family", so Abraham circumcised his infants, and even the other adult males living in his house. Abraham's faith in the promises of God, and his dedication to the commands of God, would extend, in some way, to the people who lived under his roof. It wasn't a guarantee of salvation, (Remember Esau?) but it wasn't exactly the same as being outside the family either. Instead, Isaac, and all of his children, and their children, were considered to be part of the "covenant family". Eventually, the confirming mark that they had personally understood this would be a continued reliance on God and his promises (faith). But until then, they received the sign–circumcision–that indicated that they were included in God's covenant promises.
If baptism is really a sign of the new covenant (I cannot see how it isn't), then by restricting it to adults only we are making the case that children of believing parents are not part of the covenant. To be outside the covenant is to be pagan. This is at least in part why many baptist churches practice infant dedication. We almost instinctively realize that to be the child of a person who believes in Christ is a special blessing; certainly God's promises extend to our children in some way.
But dedication doesn't go far enough, because ultimately baptism isn't about us. It's about God. The God who saves his people by sending his Son Jesus to the cross to die for our sins. The God who establishes covenants with his people and then, through his faithfulness to his own covenants, ensures their fulfillment. The God who invites us to follow him in his Grace, who saves us by his Grace, and calls us to believe. We baptize our infants because it reminds us that even when we could not respond to God, he would faithfully pursue us anyway. God's love is poured out on our children through his church, as part of the family, until they can profess with their mouth, "Jesus is mine, and I am his."
That's a blessing worth celebrating. It's grace worth signifying. That's what baptism is about. And it's why our infants receive the sign.