I bought a DR650 to chase after my son on his DR250. Both bikes are rock solid, but the lighter 250 works better in the soft stuff....650 better on the hard pack.
The DR is THE lowest seat height you can get in a DS. Drop the shock mount, raise the forks, and the short kick stand is the only part to buy. The bike is lighter and quicker than the KLR, but you need a bigger tank and wider seat to go any distance.
The Gen 1 KLR is TALL. I can tip toe, and the soft suspension will sag, the seat will give alot, but it's still TALL. The engine is solid, but benefits come w/ rejeting and you'll still get 50 mpg. I like the soft ride for any long distance, but it's TALL and HEAVY for any serious trails. Pros do it, but I don't.
The V strom is faster, smoother, and better handling ON THE ROAD. But a 19" front tire just doesn't do it on in the trails where a 21" wheel floats over obstacles. Twin engine is a rock also, but it all weighs more, and there's another reason to stay on hard surfaces.
The used market is very soft, so shop hard. Deals come after July 4th. Try to spend $3k on either DS, or $5k on a Vstrom. Get lots of FARKLES for free (used), and ride the summer away. Enjoy the shopping process, and you will be happy w/ whatever you buy.