The following four reasons give sufficient warrant to tell us that baptism should only be given to those who repent and believe and not infants.
For a fuller treatment than the summary here, see the essay on the proper recipients of baptism here.
1. The “newness” of the new covenant
Baptism is the sign of entrance into the new covenant. This is agreed to by both those who baptize infants and those who just baptize believers. Those who baptize infants argue for a close similarity between old and new covenants. Children were circumcised and entered the covenant in the old so children are baptized and enter the covenant in the new.
However, the new covenant is not the old and there are important differences. Hebrews 8:10–11 (quoting Jeremiah 31) speaks of the new covenant and says, “For this is the covenant that I will make with the house of Israel after those days, declares the Lord: I will put my laws into their minds, and write them on their hearts, and I will be their God, and they shall be my people. And they shall not teach, each one his neighbor and each one his brother, saying, ‘Know the Lord,’ for they shall all know me, from the least of them to the greatest.” This is an important difference in the new covenant. What these verses are saying is that in the new covenant, everyone knows the Lord. That is, everyone has the Spirit of God and knows God. It was not so in the old covenant. Only a remnant truly knew the Lord in the old covenant as it was a national covenant made with the nation of Israel. Paul says in Romans 9:6, “not all who are descended from Israel belong to Israel.” That is, only a remnant in the nation of Israel truly knew the Lord. Not so in the new covenant.
Why does this matter for baptism? The whole nation of Israel was included in the old covenant so the covenant sign (circumcision) was given to all born into that nation. In the new covenant the covenant sign (baptism) should only be given to those in the covenant, that is, to those who know the Lord. That is why only believers receive the covenant sign, the new covenant only includes believers (not believers and their children).
2. Those who have been baptized have what the sign signifies
Consider Galatians 3:27 which says, “For as many of you as were baptized into Christ have put on Christ.” Can such be said of infants born to believing parents? Has an infant who is baptized put on Christ? Those who baptize infants would say, surely not. All those who are baptized (received the covenant sign) have put on Christ (have what the sign signifies—union with Christ).
The same is true in Romans 6:3, “Do you not know that all of us who have been baptized into Christ Jesus were baptized into his death?” Can such be said of all baptized infants? Surely not. Or what about Colossians 2:12 which says, “having been buried with him in baptism, in which you were also raised with him through faith in the powerful working of God, who raised him from the dead.” Can this be said of baptized infants? Again, the answer is no. The Scriptures are clear, those who have been baptized, those who have received the covenant sign, have what that sign signifies—they have been united to Christ in his death and resurrection. As such, the sign should only be given to believers.
3. The command of Jesus and the apostles
Jesus commanded us in Matthew 28:20 to baptize. Jesus said, “Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit.” But baptize who? It is clear that we are to baptize disciples. Are who are Jesus’ disciples? Those who repent and believe.
The same is true in Acts 2:38 where Peter preaches and says, “Repent and be baptized.” Taking these statements it is important to realize that there is never a command given to baptize infants.
4. New Testament Examples
Lastly, there are many examples of baptism in the New Testament. Not one describes infants being baptized. When an entire household were baptized, the Scriptures are clear that it is the entire household who believed or the entire household rejoiced. That is, those who believe, who are disciples of Christ, were baptized.
It is clear that we should follow the Scriptures, uphold the new covenant sign and baptize disciples of Christ and not follow the pattern of circumcision and baptize infants. This was consistent with the nature of the old covenant but not the new covenant in Christ.
For a fuller treatment than the summary here, see the essay on the proper recipients of baptism here.