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. There is no such promise here. If you read these verses carefully you 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.
The words Paul spoke, when rendered literally, would read something like this: “You (singular) believe on the Lord Jesus Christ and you (singular) will be saved, you (singular) and your household.” Paul is proclaiming the gospel to the Philippian jailer. He tells him that if he believes in the Lord Jesus Christ as his savior, then he will be saved. This offer is not for him alone, however, but for his whole household as well. If they believe in the Lord Jesus Christ, they will be saved as well. The promise is not made that if he, as the head of the household believes, his entire household is saved by his faith.
Notice that in verse 32 Luke records that Paul and Silas preached the “word of the Lord” to him, along with his house. Verse 34 then informs us that the jailer had come to believe, along with his entire household.
The gospel is offered to the jailer, and to his household. The gospel is preached to him, and his household. The gospel is received/believed by the jailer and his entire household.
Note these words in Jeremiah 31: 29 “In those days they will not say again, ‘The fathers have eaten sour grapes, And the children’s teeth are set on edge.’ 30 “But everyone will die for his own iniquity; each man who eats the sour grapes, his teeth will be set on edge. 31 “Behold, days are coming,” declares the LORD, “when I will make a new covenant with the house of Israel and with the house of Judah,
Ezekiel 18 is perhaps even more emphatic, separating the destiny of parent and child. A godly parent may have an ungodly (unsaved) child; an ungodly parent may have a believing and faithful child. The destiny of fathers and sons is independently determined.
In 1 Corinthians chapter 7 we are told that the unbelieving mate is “sanctified” by the believing mate (7:14), but note that they are still called an unbeliever. The issue here is not the salvation of one family member due to the faith of another, but the privileged status and blessing that the unbeliever has by living in the household of the believer. God’s blessings spill over on the rest of the family, but this does not include the blessing of salvation. Paul goes on to ask in verse 16, “For how do you know, of wife, whether you will bring your husband to salvation?” Paul is saying that it is best for the believing partner to remain married to the unbeliever, if they are willing for the believer to remain, because there are certain blessings which come upon the unbeliever. He goes on to say that if the unbelieving mate does not wish to remain in that marriage, the believer is to let them go, for how can they assume that they can bring the unbeliever to faith?
All of this is to say that the Bible has a very consistent answer to your question. Every individual is lost because of their sins, and is saved if and when they trust in Jesus Christ for their salvation. This is why our Lord speaks of “hating” one’s father, mother, brothers, and sisters:
26 “If anyone comes to Me, and does not hate his own father and mother and wife and children and brothers and sisters, yes, and even his own life, he cannot be My disciple. 27 “Whoever does not carry his own cross and come after Me cannot be My disciple (Luke 14:26-27).
We must love Him, obey Him, and serve Him, above all others, including those closest to us.