1. There is never any explicit mention in the New Testament of babies (or even children) being baptized. There is lots of mention in the Old Testament of circumcision (the sign of membership in the Old Covenant) for children, but this is completely lacking when it comes to baptism in the New Testament.
2. In every verse we have about baptism, it seems that those being baptized, or having been baptized, understand themselves to be Christians—that is, they have repented of their sins and placed faith in Christ. We also see this truth in what baptism symbolizes—being united to Christ in his death and resurrection (Rom. 6). The outward symbol of beginning the Christian life should only be given to those who have actually begun the Christian life by repenting of their sins and believing on Jesus Christ for salvation. This makes good sense!

I. Gospels and Acts

a. The Bible ties baptism to conscious repentance.
i. John the Baptist’s command to be baptized is tied to his command to repent, so much so that his is called a “baptism of repentance” (Mk. 1:4, Lk. 3:3, Acts 13:24, 19:4). This requires conscious and cognitive ascent which a baby cannot produce.
ii. At the heals of Peter’s sermon at Pentecost in Acts 2, when the people ask what they should do to please God, he gives them the command to “repent and be baptized for the forgiveness of your sins” (Acts 2:38). Once again, we see these two commands intimately related—repent and be baptized. An infant isn’t capable of this conscious repentance.
b. The Bible Ties Baptism to conscious faith.
i. Acts 8:12-13: “But when they believed Philip as he preached good news…they were baptized, both men and women.” (It’s also helpful to note that this would be a natural place to mention the baptism of children since Luke has already broken it down to “men and women”. If children were baptized here, Luke’s lack of mentioning it is odd.
ii. Acts 10:43-48: “…everyone who believes in him receives forgiveness of sins through his name. While Peter was still saying these things, the Holy Spirit fell on all who heard the word…can anyone withhold water for baptizing these people?” Those who hear the Word and consciously believe it are given the Holy Spirit and it is these that are baptized.
iii. Also, Acts 16:31-34, 18:8, 19:4-5, 8:35-39,

II. Epistles

a. Rom. 6:3-4- Do you not know that all of us who have been baptized into Christ Jesus were baptized into his death? We were buried therefore with him by baptism into death, in order that, just as Christ was raised from the dead by the glory of the Father, we too might walk in newness of life. Paul goes on to say that the one who has been baptized is now “set free from sin” (6:7)—a benefit only afforded to the one who is consciously placing faith in Jesus (Rom. 3:21-26, ch. 6).
b. Col. 2:12- “…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.” Once again, Paul makes a direct connection between conscious faith and baptism. In his thinking, everyone who has been baptized is assumed to have been old enough to decide to believe the Gospel and follow Christ.
c. Gal. 3:27- “But now that faith has come, we are no longer under a guardian, for in Christ Jesus you are all sons of God, through faith. For as many of you as were baptized into Christ have put on Christ. There is neither Jew nor Greek, there is neither slave nor free, there is no male and female, for you are all one in Christ Jesus. And if you are Christ’s, then you are Abraham’s offspring, heirs according to promise.” Paul assumes that anyone who has been baptized has consciously “put on Christ” and is therefore in a self-conscious relationship with Him as Savior and Lord.
d. 1 Peter 3:21- Another example of conscious faith. The person spoken of in this passage is obviously a believer—someone who is trusting in Christ’s resurrection for his salvation.
e. Eph. 4:5- The rest of chapter 4 in Ephesians makes clear that those spoken of in this verse as being baptized are consciously trusting in Christ.
f. 1 Cor. 12:13- This verse (as do many of those in Acts cited above) links baptism with the gift of the Holy Spirit. The gift of the Holy Spirit comes from union with Christ, which only comes through conscious faith in Him (Rom. 8:9-11) and isn’t dispensed to children through the faith of parents. In fact there is nothing in the New Testament that ties Gospel-belief of parents to their children. To receive the benefits of the Gospel requires each individual believing in that Gospel for himself (Rom. 10:14-17).

3. The New Testament gives no category for someone to have half the work of God begun in their lives without the second half also. It's a package deal. "For those whom he foreknew he also predestined to be conformed to the image of his Son, in order that he might be the firstborn among many brothers. And those hwom he predestined he also called, and those whom he called he also justified, and those whom he justified he also glorified" (Rom. 8:29-30).