Sometimes, the air handler blower does not run when the condenser unit turns on. Setting the blower to ON does not help. When the blower does turn on, setting it to ON will keep it running when the condenser cycles OFF. What could cause this problem?

These things could cause this to happen, a faulty fan relay, a bad fan motor , a bad fan motor capacitor, loose connection on relay, (check capacitor first along with wire connections) these are the least expensive. Feel the motor (carefully) when it stops running to see if it is hot, it should not be to hot to touch.