Allow me to open with a quote from an article found from BibleTruths.net. (http://www.bibletruths.net/Archives/BTAR144.htm)
"I grew up in a religion that is famous for teaching "once saved, always saved." When I became identified with God's people, having obeyed the gospel and placing membership in a local church, I was presented another version of once saved, always saved: If the child has parents who are Christians, then the child "will remain faithful even though they are gone on to their own life." The argument stated is, "if you ever fall away having had Christians as parents, then they failed to do their part!"
"He must be one who manages his own household well, keeping his children under control with all dignity 5(but if a man does not know how to manage his own household, how will he take care of the church of God?)" (1 Tim. 3:4-5).
"the husband of one wife, having children who believe, not accused of dissipation or rebellion" (Titus 1:6).
Must an elder's children be Christians? This is the question that has been argued among our churches certainly without agreement. The passage of discussion is Titus 1:6, which states that an elder's managing skills are represented by, "having children who believe." Paul tells Timothy that his children are to be "under control with all dignity."
Therefore, the debate that has continues concerning the qualifications for overseer has resulted in two major views:
1) An elder must have children who are obedient (i.e. submit) to him and must behave well as part of the Christian family that is "managed" by the prospective elder (father). In other words, when they reach the age at which they become responsible for themselves, their decision to accept Christ as their Savior and obey Him is not the responsibility of their father. Yes, their father must raise them to become faithful saints, but his role as an overseer should not be decided by their choice to obey (or not obey) the Gospel.
2) An elder's children must be faithful Christians. If they are not, he is not qualified to serve as an overseer.
The passage in 1 Timothy is clearly understood to speak of the behavior of the children in the context of the "managing" or leading skills of their father, a prospective overseer of the church. It is clear that when a man doesn't demand that his children behave in a courteous and unselfish manner; when he allows them to become unruly, discourteous and unruly, he is not fit to lead the people of God.
However, Titus uses these words concerning the overseer, "having children who believe." Is this to mean that if his children must remain Christians the entire time that he is an elder? Or could this again refer to the faithfulness of the children to their father and his authority?
Someone has rightly pointed out that these two letters (Timothy and Titus) were written to two different men in two different places, yet include the qualifications for the same "work" in the church. Would the two then not be consistent with one another? Whatever was expected of the overseer's children in Crete (Titus) would also be expected if Timothy appointed elders. But nowhere in 1 Timothy do we find the requirement that a man's children be faithful Christians.
It would seem reasonable then to conclude that the word "believe" (pistos) in Titus 1:6 refers to the attitude of children toward their father (and therefore the church) before it could be concluded that they were responsible for the destiny of their own souls. As the quote at the beginning of the article points out, the salvation of a shepherd's children is no different than with any man (or woman) . . . . They must be responsible for their own salvation, having been taught the way of the Lord as children. This is not to trivialize the importance of the overseer teaching and managing his household in a manner that will produce faithful children of God. And that seems to be precisely what Paul is saying in Titus 1:6.