Well I wouldn't panic and think he's dying. Bettas are unlike most fish in that they can breath air. So, most fish swim around because it's easier for them to send water through their gills, but bettas don't necessarily have to do that. On the other hand, I wouldn't say he's healthy. Healthy bettas are usually cruising around the top of the tank making a bubble nest. You can use the bubbles as a really good indicator as to if he's healthy of not. Try changing his water a little more often and see if that gives him a little jump start.