Whether or not the Jailer was asking for a worldly salvation, Paul was offering him an eternal one. And in the end the jailer believed and was baptized. Rather than believing in Caesar’s power, his superior’s control, or any of the many false gods of his day, Paul told him to put his faith in Jesus.
“Believe in the Lord Jesus Christ and you will be saved,” is still a good message for us. We can look to false gods, fear, money, sex, power, or a myriad of other demigods. None of those will give life, much less eternal salvation. The question is whether or not we fully understand the depth of our need for Jesus, or the full meaning of being saved. When we put our faith in Jesus we will be saved. And there’s no question about that.