Paul and Silas" love for him, in contrast to the hatred they had received from the magistrates, the police, and the jailer, transformed the jailer"s attitude. Apparently the jailer had heard the gospel from Paul and Silas previously, or had at least heard what they were preaching (cf. Acts 16:17), but had hardened his heart against it ( Acts 16:24). Now, because of his brush with death, he humbled himself and asked how he could be saved. Another possibility is that the jailer only wanted deliverance from his physical danger. [source][source][source]
". . . if these were the jailer"s exact words they probably meant: "How can I be saved from the consequences of having ill-treated two obviously powerful magicians?" Paul uses the question as an opening for his Gospel message ( Acts 16:31)." [1][source]
"The earthquake has presented him with irrefutable evidence that God is at work with Paul"s group. He wants to know whatever more Paul can offer. Is there a way to escape God"s reaction to the injustice in which the jailer has played a role? In the face of this evidence, the jailer does not want to be found on the opposing side." [2][source]