Select a drill bit that is the same size as the screw that is stripped. Install the drill bit into the chuck at the end of the drill. Some drills have a key that is used to loosen and tighten the chuck, and some chucks are operated by hand.
Hold the drill so that the bit is completely vertical to the screw in the handle. Squeeze the trigger and guide the drill into the cavity of the faucet handle as it drills through the screw. Drill out the screw until you can remove the handle.
Stuff a piece of steel wool into the screw hole at the top of the valve stem with a Phillips screwdriver. Turn the steel wool in the screw hole with the screwdriver to remove any residual particles of steel from the screw hole.
Spray lubricant into the screw hole in the top of the valve stem to help prepare it for a new screw when the handle is reassembled.