I am afraid that this is one of those Scriptures that when taken out of context has caused a lot of grief and pain.
Some wish to prove from this text that when the head of a family comes to faith, he can be assured that his whole family will follow him in faith however......there is no such promise here.!!!
When we read all of the verses carefully we will see that the offer of salvation by faith in Jesus was offered to the jailer, and to his whole household. The message of salvation was proclaimed to the jailer, and his whole household. The message of salvation was believed by the jailer, and his whole household. And as the jailer professed his faith by being baptized, so did the others in his household. The offer was to the jailer and all of his household. The acceptance of this offer was by the jailer and his whole household, but there is no guarantee that this will always be so.
We are all saved by faith in the Lord Jesus Christ ONE AT A TIME. There is NO such thing as household salvation!