This is a question that must draw believers to a want -to-know answer. I personally have witnessed to my mother, sister, father, friends, aunts and uncles and workmates. And as a believer since 1997, I have seen few of them become faithful followers of Christ. I have seen a good number of others come to saving faith, and myself having played an instrumental role in that. But my own family, not so. But, this does not overwhelm me. Even though this is discouraging, I know, as domingoosable has pointed out, we are instruments chosen by God to reach them. God saves them, we make disciples.
I believe that the Acts 16 passage is specifically about who would believe. Notice that the four times the whole household is mentioned, that same household did something the same as the jailer. “Believe in the Lord Jesus”, and “they spoke the word of the Lord to…” (they all heard the gospel), and “immediately he was baptized, he and his household”, and “rejoiced greatly, having believed in God with his whole household”.
Belief in Jesus as Lord, accepting the truth about our sin and need for salvation through Christ’s sacrifice, all that is part of getting saved. Just because we might become a believer, that will not determine that our families will get saved. However, we become the fixed gospel on display in every person in our influence. We only have to live what we believe and every unbeliever in our family, friends, workmates and schools, neighbourhoods and influence, if we are living Jesus out, they will have every reason possible to want to know the God of saving grace.
I never give up hope on my unbelieving family. I sense God is wanting my passion for their salvation to make me a more sincere follower of Him. Until He takes the breath of life from them, I will continue to pray believing that one day, they will embrace salvation. And if they do, I will know it. Because where God’s Spirit is working, so there also will be the fruit of His presence.
Hope this helps.
“They said, “Believe in the Lord Jesus, and you will be saved, you and your household.” And they spoke the word of the Lord to him together with all who were in his house. And he took them that very hour of the night and washed their wounds, and immediately he was baptized, he and all his household .And he brought them into his house and set food before them, and rejoiced greatly, having believed in God with his whole household.” Acts 16:31-34 NASB