This is the same belief that the man who I met in Fort Mc Murray area had. He told me that there was actually a baby in the jailer's household. I told him he was wrong and that the jailer had three children: one was 12, another was 15, and the oldest 18. He was surprised and asked me where I got all that! I asked him the same thing, where did he take the baby from the verse! He could not point it out.
How do we know, then, if there was a baby in the household or none? Let us read a few verses before and after verse 33.
Acts 16: 30-34 (NLT):
30 Then he (the jailer) brought them out and asked, “Sirs, what must I do to be saved?”
31 They replied, “Believe in the Lord Jesus and you will be saved, along with everyone in your household.” 32 And they shared the word of the Lord with him and with all who lived in his household.33 Even at that hour of the night, the jailer cared for them and washed their wounds. Then he and everyone in his household were immediately baptized. 34 He brought them into his house and set a meal before them, and he and his entire household rejoiced because they all believed in God.
Babies do not have the ability to believe because they are not yet developed. They are still in the innocence stage. Later, as they grow older they will realize and understand that they are sinners like anybody else. Hence, they can not believe in the same way the jailer's household did. They do not have a concept of sin and the need of a Savior--the Lord Jesus (verse 31). Those who can have the FAITH, that Paul was talking about, are those who are old enough to understand their need of salvation. "Believe in the Lord Jesus and you will be saved..." This call do not apply for babies.
Likewise, we can read that Paul "shared the word of the Lord..." (verse 32) to them and "all believed in God" (verse 34). Sharing is a communication between and among people who can exchange ideas and thoughts. It is a process of giving and receiving. Furthermore, to believe is to look upon and accept the word of the Lord as true, and to be confident about it. Babies can not "share" and they can not "believe" the word of God.
Moreover, as Christians, we know that part of believing in the Lord Jesus is a personal confession of one's faith (belief). Only those who receive salvation are entitled for a baptism. So in the case of the jailer's household, only those who confess their belief were baptized. Since babies can not confess their faith/belief, how can they be baptized? Because everyone (nobody left behind) in the jailer's household (all of them believed) were baptized, it is proper to say that there was no baby in his household.
Nobody knows the exact age of the members of the jailer's household and how many people comprise it. But by reading the verses above, we know that everyone in the household were sinners who believed in the Lord Jesus and were saved by grace.