If the door handle has no spindle through the door (they usually do) just screw the affected handle higher than your son can reach - he won't be able to swing on it then.
If the handles are on a spindle buy a new, stout set and fix them on solidly.
Constant swinging will, over time, put great strain on the screws holding the top hinge - the boy might get a surprise when the door falls on him - ask him to wear a cycle helmet for safer swinging.
Children are such a blessing...
You might also consider using wooden plugs in the screw holes before reattaching the handle next time - will give the screws something to bite on.
Good luck...
Peccavi, June 2009