What to do if betta fish is barely breathing at the bottom of tank?

Your betta is probably doing this because your water quality is not within healthy limits. If you haven't been doing consistent weekly water changes then the water could contain extremely high amounts of ammonia and nitrite or even nitrate. Water changes are needed regularly and if not done ammonia molecules which are bonded to oxygen molecules by nitrosomas bacteria will buildup and make it extremely hard for the fish to breath. Thus explaining the lethargic behavior. This can result in irreversible gill damage. Do a water quality test to check if your waters within safe perimeters. If not do several large 50-75% water changes until the ammonia, nitrite and/or nitrate levels drop.

Another reason why the fish could be acting this way is if it is sick. Usually an internal bacterial infection causes a fish to lose the ability to maintain proper buoyancy because the swim bladder may fill with fluid. Other than bacteria being the problem your fish may be sitting at the bottom because its being bombarded by parasites (such as flukes).

Treatment:

Try doing several large water changes, treat the tank for parasites using an anti-parasitic medication and feed the fish an antibacterial medicated food.