1 Corinthians 7:14

For the unbelieving husband is sanctified by the wife, and the unbelieving wife is sanctified by the husband: else were your children unclean; but now are they holy (KJV)

Then: Everybody has a different opinion of Paul's reasoning for this verse, early church fathers onward. Separating one thought from another is like looking down into a top-loading washing machine as the clothes swish from one spot to another in the water. Nothing stays put. The issue is "How can lack of faith be holy?" The only given is the center pole - the presence of the believer makes the unbelieving husband and children holy. As God provides grace to the believer, anyone living nearby also benefits from that grace and goodness. Although readers can observe that this verse contradicts Paul's previous requirement that a believer flee all immoral liasons (Corinthian infidelities), the believer's place before the Lord makes the difference. Sex within marriage is lawfully designed by God while sex outside of marriage is not. The second is condemned, that is, not capable of ever being holy before him. Citing Collins, Baker (p.104) brings in Paul's knowledge of Jewish heritage, whereby proselytes and their children were fully accepted as holy in the community. That's one comparison.

Now: Personally, I believe sanctity comes from God's mercy. When Lot lived in Sodom and Gomorrah, the Lord promised Abraham that if more than ten righteous people could be found, he would save the cities. When that proved impossible, the Lord sent Lot and his family away and the land was destroyed. In the same vein, Jesus taught that a little yeast will leaven the whole, salt maintaining its saltness has value, and light can overcome the darkness. In a marriage, the righteous believing spouse who receives God's mercy and favor makes it possible for all members of the family to live in that same atmosphere. He or she is the leaven, the salt, or the light. It does not mean that in the Day of Judgement, non-believing members of the family will have eternal life. It only means they are temporarily blessed with a place of God's protection.

Note: In conversation with a friend, I learned that I had ignored the Christians's prayers on behalf of the family. That's an additional pathway to grace and mercy for unbelieving spouse and children.

Extra source: Cornerstone Biblical Commentary 1 Corinthians (William Baker), 2 Corinthians (Ralph Martin & Carl Toney), ed. by Philip Comfort. Carol Stream: Tyndale House Publishers, c2006.