How is a Christian to deal with a spouse who has no interest in sex?

This is a very complex question with several different possible things to address, thus, I may only make general comments.

As most are aware, a husband and wife should not "defraud" one another accept by mutual consent (1 Cor. 7:5). The King James English uses the word "defraud" and I use it here because it brings up an interesting point: a theft is taking place. However, your question is not whether this is wrong but rather what should one do about it.

First of all, one must not jump to conclusions about why this is taking place. Having no interest is sex may be the result of several things happening in a person's life. There may be emotional struggles with past abuse. There may be a present issue with relationship with God. There may be a medical impairment which is causing a lack of desire. Childbirth, particularly when recovery is prolonged by postpartum depression, may hamper a woman's arousal. Various prescription drugs may adversely affect sexual urges in males and females alike. There may also be unresolved problems within the marriage (Eph. 4:6). Or, as everyone's mind seems to jump to first, there may be another person. Still, it is best not to jump to conclusions (Rom. 12:1-2; Heb. 12:1-4; Phil. 4:4-9; cf. Prov. 3:5; 4:23).

When dealing with such a situation, prayer and the Word of God is essential. Just as breathing is to our survival as humans, prayer is the Christian's life force in this present existence (Psa. 66:19-20; 69:3; 102:1; 1 Thess. 5:17). Individual communion with God is extremely important. It is part of the armor of God in Ephesians 6:10-18, and we cannot stand successfully in the midst of trials without it. In addition, the Word of God gives us wisdom to act wisely (Prov. 3:1, 3, 5; 4:21; 10:8; 12:8; 15:28; 16:9, 23; 18:5; 19:21). As one also seeks godly counsel (Ps 16:7; Prov. 8:14; 11:14; 12:15; 15:22; 19:20-21; 20:5), decisions will have to be made, and communion with God will aid in making such decisions. God may teach many things during such a trial, one of which is not to let this trial turn into a temptation (1 Cor. 7:5).

Lastly, I recommend seeking the counsel of "a" pastor with counseling abilities, keeping in mind not all pastors are gifted in this area. "A" pastor may not necessarily be "your" pastor. One should submit to "a" pastor, but a loving and wise pastor should know his abilities and giftedness or lack thereof (1 Cor. 12:4-6). It may be better to go "out of town" and away from prying ears to discuss such situations. Regardless, what is important is that such situations be shared and discussed with someone who is competent to counsel and who will listen and be able to give biblical advice.