Mixed Marriages Legitimate
1Cor 7:12-14 To the rest I say this (I, not the Lord): If any brother has a wife who is not a believer and she is willing to live with him, he must not divorce her. And if a woman has a husband who is not a believer and he is willing to live with her, she must not divorce him. For the unbelieving husband has been sanctified through his wife, and the unbelieving wife has been sanctified through her believing husband. Otherwise your children would be unclean, but as it is, they are holy.
Here God legitimizes non-Christian marriages. Not that a Christian should decide to marry a non-Christian. "Do not be yoked together with unbelievers. For what do righteousness and wickedness have in common? Or what fellowship can light have with darkness? What harmony is there between Christ and Belial? What does a believer have in common with an unbeliever." 2Cor 6:14,15 and likewise he advises widows at the end of this chapter, "she is free to marry anyone she wishes, but he must belong to the Lord." 1Cor 7:39
His comment regarding the children is related to what God said in Malachi, "Has not the LORD made them one? In flesh and spirit they are his. And why one? Because he was seeking godly offspring. So guard yourself in your spirit, and do not break faith with the wife of your youth." Mal 2:15, which I believe is a rebuke to people like Nehemiah whom, due to prejudice, was exhorting men to divorce their wives, and by doing so set an ungodly example for their children. One spouse being a believer is sufficient to influences their kids into the kingdom. But he is not saying that such children are automatically saved, nor their spouse. Afterall doesn't he say a couple of verses from now, "How do you know, wife, whether you will save your husband? Or, how do you know, husband, whether you will save your wife?" One is not to presume one's whole family is saved simply because one member of the family becomes a Christian. But don't underestimate your ability to influence other members of your family into the kingdom.