The current tires are the knobby stock tires that came with the bike and they do great on the Mountain. Unfortunately, they scare the hell outta me on the road at high speeds (the like to pull) and in any slight wind. I'm fine (even in the wind) if I stay below 45 mph, but above that - it get's frightening.
As mentioned, check your tire pressure..
I, and ALL the KLR riders I know have no problems riding a KLR at highway to "going to jail, boy" speeds. I often ride all weekend at 65-70 with zero issues due to speed, wind, surface, etc..
While the KLR is taller and a bit more affected by wind, unless it's 20mph or more crosswind, I don't even notice it.. And the FASTER you go, the less wind affects you since the headwind of speed is greater than the crosswind as you go faster.
So, check your current tire pressures (as mentioned) and kick around running road pressures when on the road (32psi F and R is a good place to start) I actually run 34f and 37r on my KLR. I air down to 20 for extended off road playing and then carefully ride to the next gas station and air back up when done playing.
As far as what tire for everything, there's no such beast. All tires will be a compromise. A great street tire will last a long time and be so/so on the dirt or loose stuff, and a agressive dirt tire will be squirmy on the street, and will go away within a couple thousand miles.
The trick is to find a tire that does everything fairly well, and still has a decent life span. That said, there's several choices in that catagory, and it'll depend on if you want to pay $100 for a set, or $300 for a set..
Before jumping into this, let's discuss it a bit more with your input.
First, how much riding have you done, and let us in on your tire pressures.