Jesus, after all, came to mend that which was broken. Not just broken individuals, but broken marriages, broken homes, broken relationships. His aim is not to see just isolated individuals who have a relationship with God, but families, loving and worshipping Him together.
When I seek to explain household salvation to people, they usually say something like "But it's not automatic, is it?" or "But people have to make their own decision, don't they?" Of course they do. A decision to follow Jesus is a personal choice.
But salvation is a gift. God can give it to whomsoever He will. The book of Ephesians says "For by grace you have been saved through faith, and that not of yourselves; it is the gift of God, not of works, lest anyone should boast." No man can come to Christ except the Father draw him. If that is the case, then even the willingness to accept Christ comes from Him.
Plainly not all families are united in Christ. So if "you and your household" is a promise, can it be that there are conditions attached to the promise?
I believe that there are. First, it is necessary to believe the promise. Second, it is necessary to claim the promise. Third, it is necessary for the believer to remain faithful. Clearly, it is not reasonable, having accepted Christ, to live all sorts of a life and expect one's household to come to faith.
I have heard it suggested from the pulpit that Paul and Silas said "Believe on the Lord Jesus Christ, and you will be saved, you and your household" to the Philippian jailor as a result of a word of knowledge specifically for the Philippian jailor, and that believers shouldn't claim that promise lest they be disappointed.
But I want to suggest to you, in view of the Scriptures I pointed out here, that we should expect whole families to be saved, and that in "you and your household" there is a principle that God is prepared to stand by.
The Chinese Christian Watchman Nee believed in household salvation. He tells in one of his books how, when he visited England, he called to see George Cutting, a well known evangelical at that time, who also believed in household salvation. George Cutting, then an old man, had more than 80 in his family - sons, daughters-in-law, grandchildren and great-grandchildren. Every one was saved.
Is that coincidence, or what?
 Watchman Nee. The Good Confession, vol 2. New York: Christian Fellowship Publishers, 1973, p113.