Discuss it. Dad's often get left out of the pregnancy process. Pregnancy is an abstract experience for men where as it is very concrete for women. Discuss his reasons for not wanting to participate but be careful not to judge or be critical. Have an open discussion. Some people find classes fun and education. Others prefer a hands off experience b/c classes create anxiety. Ask him his reasons.
Well. If you get an epidural, the birthing classes may not be necessary. The reasons he doesn't want to go can be endless but don't put too much pressure or read too much into it (ie "he wouldn't even go to birthing classes"); it's gonna add unnecessary stress at a time when relationship stress goes up incredibly. If deeper problems seek therapy for yourself or both of you; clearing out stressors good.