Here is the passage in question:
1 Corinthians 7:36-38 But if any man think that he behaveth himself uncomely toward his virgin, if she pass the flower of her age, and need so require, let him do what he will, he sinneth not: let them marry. Nevertheless he that standeth stedfast in his heart, having no necessity, but hath power over his own will, and hath so decreed in his heart that he will keep his virgin, doeth well. So then he that giveth her in marriage doeth well; but he that giveth her not in marriage doeth better.
I believe that it it our culture that makes this passage so difficult. The key is to identify the "man" and "his virgin." The key is found in a couple of phrases. The man is the one who will "let them marry" (v.36). He is also the one who either "giveth her in marriage" or "giveth her not in marriage" (v.38). The man in question is obviously a father and "his virgin" would be his daughter.
In 1 Corinthians 7, Paul teaches about marriage. In one large section of the chapter, he is giving guidance concerning marriage in a particular situation. He calls this situation "the present distress" (v.26) and much of what he says is geared to this. That is, the Corinthians were going through a time of special trial, probably a time of persecution. Paul understood that marriage in such times can create more sorrow than joy and he wisely warned the Corinthians of these dangers. However, he never denied them the ability to marry. He only warned them of the problems of marriage in such times.
Our passage describes a father who has just been told that it is best that his daughter not marry. However, he is seeing her get older and is fearful that she will miss out on her best opportunities for a good husband. Paul looks at two different situations. In the one (v.37), the father can decree in his heart and be at peace with keeping his virgin daughter unmarried. For the other father (v.36), the idea of keeping his daughter unmarried is disturbing. He wants to please God, but the importance of giving his daughter in marriage is part of his culture and inner makeup. It is very important to him. To this father, Paul tells him to go ahead and give her in marriage. In Paul's conclusion (v.38), both actions are good though not giving her in marriage is better.
In order to understand this passage, we must comprehend two concepts. First, Paul is dealing with a special time of distress (1 Corinthians 7:26). His teaching is not strictly applicable to every situation. Second, the passage deals with a cultural difference that is foreign to recent western peoples. In this passage, Paul is concerned about the struggle that the father has over whether or not to give his daughter in marriage. We wonder about the daughter and her feelings. This points out an important difference. The giving of the daughter in marriage was not a quaint part of the wedding ceremony in Bible times. It was a reality. No permission; no marriage.
This is not the place to discuss the right or wrong of fathers giving their daughters in marriage. However, it is found throughout scripture. The first reference is found in Genesis 34:8-9 and deals with Dinah the daughter of Jacob. It was taken seriously in America until the last couple of generations. It is still taken seriously in other parts of the world. Perhaps a godly daughter should at least take getting her father's permission seriously. Certainly, God would bless those daughters who determined not to marry outside the will of their fathers.