I used to have a KLR. Bought it for exactly the same reason. They are great bikes for what you want to do, but...I'm considering doing a multi-state ride through AZ, CO, WY and the probably back down via MT, ID and UT. Throwing in a lot of dirt; mostly forest road but some moderate stuff with some of the CO passes. Could the KLR be an appropriate bike for this?
Once you get the bug to tackle more difficult terrain, the KLR will make you work a lot harder, which is fine as long as you are comfortable with that, because the bike can handle it.
When I got mine, I was very rusty and out of shape. I had the habit of putting it in places that made me work really hard, but still enjoyed it. That being said, if I was to do it all over again, I probably should have got a DR 650 to start out with due to my dirt preference and not caring as much about highway comfort in stock form. Now I ride an enduro, get blown around a bit on the highway, chew thru tires faster than gas, spend $65 to change the oil myself, and love every minute of it