"So they said, 'Believe on the Lord Jesus Christ, and you will be saved, you and your household.' Then they spoke the word of the Lord to him and to all who were in his house. And he took them the same hour of the night and washed their stripes. And immediately he and all his family were baptized. Now when he had brought them into his house, he set food before them; and he rejoiced, having believed in God with all his household" (Acts 16:31-34).
In order to properly understand any portion of Scripture, one must consider it in context (i.e., the immediate context in which the verses are found as well as the wider context of the entirety of God's word). Sadly, the context of this passage of Scripture has been frequently abused in an attempt to support a man-made doctrine. Many sermons on salvation have been preached in the denominational world that focus on Acts 16:30,31 to the neglect of verses 32-34 and the rest of the New Testament! Great emphasis is placed upon the jailer's sincere question ("What must I do to be saved?") and the opening response by Paul and Silas ("Believe on the Lord Jesus Christ, and you will be saved"). Then, denominational teachers will often proceed to claim that salvation is by faith only and that one just needs to believe in Jesus to be forgiven. Friends, this is a perversion and twisting of Scripture!
It is absolutely true that one must "believe on the Lord Jesus Christ" in order to be saved. There is no question about that. But, there is great disagreement over what it actually means to "believe on the Lord Jesus Christ." Is believing on Christ something one can accomplish purely in his mind (e.g., mentally assenting to the truths about Jesus that the Bible teaches)? Or, is there more involved in believing on Christ? God's word has the answer! An objective look at this immediate context (as well as the entire book of Acts) will answer the question definitively.
When the jailer asked what he needed to do to be saved, he had very little knowledge about Jesus and God's plan of redemption. He knew he was in need spiritually, but he may not have known much more than that. Paul and Silas initially respond with a summary statement of how he could be saved - "Believe on the Lord Jesus Christ, and you will be saved, you and your household." But, they don't stop there (and neither should modern-day preachers who use this passage to teach on salvation)! Paul and Silas "spoke the word of the Lord to him" and to his household (likely when the jailer brought the men out, he took them into his residence where his family was, which would have been near or even attached to the jail). In other words, these faithful gospel preachers explain in detail what it actually meant to "believe on the Lord Jesus Christ." They certainly did not simply teach the necessity of mentally acknowledging certain facts to be true.
In addition to teaching the jailer about Jesus and the importance of faith in Him, it is implied that he learned about the need for repentance (cf. Luke 13:3; Acts 17:30,31). His actions here show a penitent spirit, specifically when he washes their bloody wounds after hearing them preach. Immediately afterward he and his family were baptized (i.e., immersed in water for the forgiveness of their sins; cf. Mark 16:16; Acts 2:38; 22:16). Carefully note in Acts 16:34 that Luke does not state that the jailer "accepted Jesus into his heart and was saved." Luke recorded that the jailer "rejoiced having believed in God." To believe in God is undoubtedly used here as a summary term (as it was in 16:31) to include all that the jailer did (i.e., heard the gospel, developed faith, repented, and was baptized for the remission of his sins). To do all those things is to "believe on the Lord Jesus Christ." One who has done such has good reason to rejoice, for his sins have been washed away by the blood of Jesus (contacted through baptism) and he now stands righteous before God!
Two side notes are worth observing here: