By far the most frequently used argument to justify infant baptism is the fact that the New Testament talks about “household baptisms,” and some households include infants. Let’s examine this argument in detail.
First notice that households (“the inmates of a house, all persons forming one family, a household” – Thayer) do not always contain infants, so this argument is just conjecture. As a matter of fact, the great majority of households do not contain an infant. Proof must be conclusive, not just a possibility.
One household that is often used as a proof case is the household of Cornelius. But notice that Cornelius called together his kinsmen and friends (Acts 10:24) to hear Peter preach, and the ones that were baptized were those who “feared God" (10:2), "believed on … Jesus”, and repented (11:17-18). Doesn’t sound much like infants, does it?
Regarding the household of Lydia in Acts 16:15, she was a busy merchant woman (16:14), and not even married as far as we know. Knowing this, how could one use speculation that she had infant children who were baptized as evidence for infant baptism?
In that same chapter we have the baptism of the Philippian jailor “and all his” in Acts 16:33. But verse 34 tells us “all his house” believed in God. Believing is not possible for an infant, right?
It is true the household of Stephanas was baptized (I Corinthians 1:16), but I Corinthians 16:15 also lets us know that his house was “addicted … to the ministry of the saints.” I don’t think infants are capable of being addicted to the ministry, do you?
The last case I remember being brought up is the house of Crispus. But Acts 18:8 says Crispus “believed on the Lord with all his house,” so we know infant children aren’t included in the number that were baptized because the same ones that were baptized were believers – ruling out infants of course.
There is a lesson to be learned about Bible proof from this topic that we already recognize in the areas of science and law. Conclusive evidence must be just that – conclusive, not just a maybe, might, or possibility. The bottom line is – there is no mention in the Bible of infants being baptized, ever.