Abraham was circumcised following his trust in God’s covenant promise. His male seed were circumcised before their existential trust.
In the Old Testament, circumcision is not an either/or matter. One might say that Abraham became a believer, and therefore, having become a believer, he was circumcised. His children were born within that covenant family, and as a sign of that covenant, they were circumcised with a view toward their responding to the covenant promise.
The new covenant follows the same pattern. Those who come from outside the new covenant community, who are converted to Jesus Christ, will be baptized as converts on profession of their faith. Further, in the same way as was true with Abraham, because the covenant promise is not just given to them but to their children, those children also receive the covenant sign. On that point, there is no difference between the way circumcision was administered then and the way baptism is administered today.
As Paul points out in Romans 4:11–12, circumcision was a sign of the gracious, saving righteousness of God’s covenant promise, which Abraham received by faith. His circumcision was not a sign of his faith but a sign of God’s promise. Therefore, it could be given to Abraham, who trusted that promise. In other words, circumcision was a sign of the promise he had already trusted. Similarly, his male children could receive circumcision because it was a sign of the promise and not a sign of faith. As such, their circumcision called them to trust in the promise.
The pattern is: I come to faith in Jesus Christ, I am baptized, and my children are baptized even though they may not have come to faith yet. Baptism is not a sign of my faith; it is a sign of the gospel of Jesus Christ. So, my children are also baptized, and that baptism becomes a visible presentation of the gospel that summons them to faith and repentance.
There are obvious differences between circumcision and baptism. However, there is also obvious continuity in God’s administration of the covenant sign and the covenant promise.