It looks like an Allen wrench is the right tool, but it also looks like the hex is all chewed up from someone using the wrong size wrench. It's also possible that it may need a metric size. If you can't get it out with the correct wrench, you could try to use needle nose visegrips as the set screw appears to protrude a bit. If that doesn't work, your only option would be to drill it out and retap the hole. Good luck...

Thanks for the suggestions.

I have 3 doors like this. The other door with the same screw is flush with the handle. I will get some ease outs and try them.

Check a metric Allen wrench

I've never had much luck with an easy out, although it is worth a try. I would attempt using the tightest fitting Allen wrench, standard or metric, first. Then I would try the vise grip; then the easy out; then drilling the hole deeper and trying the easy out again. If all that fails, then you'll have to completely drill the Allen screw out and re-tap the hole.

Holefully the set screws in the other doors are not as abused as this one....

As true with most tools,you must follow instructions while useing ease outs. They have removed many engine head bolts for me,allen screws that are not rusted are child's play.

for these small set screws OP maybe able to get away with a set of reverse drill bits and an electric drill. Drill it out and E-Z out it at the same time.

If you are going to replace the assembly, saw the handle off with a hacksaw or reciprocating saw.

Get away from here Mac. Magin waht our cussamu gonna think if she catch us sawing off old dough knobs. You jess want a good laff on us foe looking silly don't you?.

I ended up using speed outs.

http://www.homedepot.com/p/Ontel-Speed-Out-Speed-Out-Screw-Extractor-4-Piece-1000264/205110987

The actual door lock was quite complex to take out. YouTube saved me here.

Thank you for all the suggestions.