There´s never an incident of a baby being baptized in any of those households"”it never identifies them. "Households" simply mean"”could mean "family, could mean "servants" who were a part of that household.
1. Cornelius´ house"”Acts 10. The gospel was preached by Peter, Cornelius heard it"¦it says, "They all heard the Word"¦they believed it"¦the Spirit fell"¦they were all baptized." All heard, all believed, the Spirit came on all, they were all baptized.
2. In the jailer´s house"”Acts 16 is the next one"¦Philippian jailer. Paul, you remember, gave him the gospel, it says, "All heard the gospel"¦all were baptized."
3. Chapter 18, it was in the house of Crispus, "All believed"¦all were baptized."
The other two occur in I Corinthians. The other two are the account of Lydia and Stephanas"”Lydia is in the book of Acts.
4. But, in the case of Lydia, it´s the same thing. We must understand the same thing must have occurred"”they heard, they believed, they were baptized.
5. Stephanas: They heard, they believed, they were baptized.
I mean, it´s all basically the same pattern. They all hear the gospel, they all believe, they all receive the Spirit, they all are baptized. That excludes infants because infants can´t hear and believe. The "household" then is defined"”it is defined as "those capable of hearing, understanding, believing." That´s the definition of the "household."
In Stephanas´ household, which is in I Corinthians, chapter 1, "All who were baptized," it says, "All who were baptized were devoted to the ministry of the saints." Babies can´t be devoted to the ministry of the saints. It says, "All who were baptized were helping in the spiritual work of the church." It´s impossible for infants.
In the case of Lydia, in Acts, "her heart was opened when she heard the gospel. The gospel was preached and her heart was opened," it says. So, we understood she heard the gospel, she believed"¦others must have heard the gospel, their hearts were opened, and they believed and they were baptized. By the way, to assume there were children in the house is maybe stretching it since, apparently, she had no husband. She, apparently, was a single person.
In John 4, in verse 53, it says about a nobleman"”you know, whom Jesus talked with and He healed his son"”it says about that man, "He himself believed and his whole household." They all believed. Household belief, then household baptism. Where there is no faith, there is no baptism.
http://www.biblebb.com/files/MAC/INFBAP.HTM