Act 16:27 When the jailer woke up and saw the prison doors wide open, he drew his sword and was about to kill himself, since he thought the prisoners had escaped.
Act 16:28 But Paul shouted in a loud voice, "Don't hurt yourself, because we are all here!"
Act 16:29 The jailer asked for torches and rushed inside. Trembling as he knelt in front of Paul and Silas,
Act 16:30 he took them outside and asked, "Sirs, what must I do to be saved?"
Act 16:31 They answered, "Believe on the Lord Jesus, and you and your family will be saved."
Act 16:32 Then they spoke the word of the Lord to him and everyone in his home.
Act 16:33 At that hour of the night he took them and washed their wounds. Then he and his entire family were baptized immediately.
The natural man, before he has been regenerated, would not have responded to Paul in the manner that the Jailer responded.
Before the natural man has been regenerated he has a stony heart, as indicated in the men that stoned Stephen to death for preaching the same sermon that Peter preached on the day of pentecost.
In Stephen's case the men that stoned Stephen were "CUT TO THE HEART", and the Jews, at pentecost, who had had their stony heart changed to a fleshy heart in regeneration, responded in the same manner as the Jailer, as they were "PRICKED IN THEIR HEART" and ask "what must we do".
When considering your text of Acts 16, The word "saved" in verse 30 & 31, according to the Greek meaning, means "to save, i.e, deliver or protect.
The scriptures teach that there is an eternal deliverance, and there is a deliverance that the regenerated person receives, by their good works, as he sojourns here in this world, that is not eternal.
By misinterpreting this, many of God's well intended regenerated children, tend to believe that there is something they have to do to be delivered eternally.
2 Tim 2:13, If we believe not, yet he (Jesus) abideth faithful, he (Jesus) cannot deny himself. Eph 1:4, According as he has chosen us "in him" before the foundation of the world.