Either bike would be good.
The KLR a bit better for comfort load carrying and runs well on crappy fuel.
The DR has a great reputation also.
Depends on which one you can get a better deal on.
In the USA there are lot's of deal on KLR's. The DR650's tend to be more expensive.
Get a clean used KLR, make sure the doohickey was replaced, the valves checked, subframe bolts upgraded, grease the swingarm, and you're good to go 20,000 miles.
All you'll need along the way are tyres(the KLR tyres are a common size and fairly cheap), Brake pads, change the oil and filter a few times along the way, and you'll probably have to change sprockets and the chain at some point.
1995 R1100GS (battle scarred Asian)
2006 KLR685 (the America's tool, 42K and counting)