I've owned both: 3 KLR's, and now my first DR.
The KLR is slightly smoother-running, though I had to gear it taller for freeway riding. It also seems to have more touring gear available: bags, highway pegs, panniers, etc.
Everyone talks about shorter gearing for the dirt, but neither bike is a real dirt bike (60-80 lbs too heavy), so that makes no sense to me.
The DR is easier to lower, and you can get a big tank for it. (I just bought an IMS 5.5 gallon tank for $120 incl. shipping). And it has much better brakes!
They will both run on regular gas, if that's important to you.
Right now I'm partial to the DR, but I really liked the KLR too--it was very reliable for me.
These days it seems like you get a bit more DR for your money: they're a little cheaper for the same age and mileage, but then you have to buy a bigger tank for them, so that comes out about even.
Try getting a ride on both bikes. The DR is a little lighter, and being easily lowered, that could be the clincher for you. Good luck!