FWIW, here's my standard "wobble post", though I'd note that it deals with the common high speed wobble more than anything else;
Way too many people think that addressing the symptoms by dealing with handguards, fenders, fork braces, etc. are the answer rather than dealing with the real issue which is related to suspension setup and loading. I'm not convinced that the KLR is any more susceptible to instability than any other bike with long travel, lightly damped suspension and the Owner's have a propensity for severe and uneven loading.
There are some problems that need to be checked;
- bad/lose head bearings
- condition of wheel bearings and suspension bushings
- wheel and tire condition and appropriate tire pressures.
.....beyond that, It's settings;
- proper sag settings and adequate damping
- proper bike loading
- avoiding inappropriately un-aerodynamic loads
addressing the symptoms rather than the cause can help but IMO shouldn't be done until all the aforementioned items are checked and corrected if necessary. Nonetheless these can help stability;
- fork brace
- smaller fender or lowered fender (I use a polisport as I hate both the supermoto and low mounted fenders)
- consider tank bags instead of putting everything in huge panniers which affects both weight loading and aerodynamics.
My 2001 had some high speed issues that went away as soon as the sag was set properly....and after my Cogent suspension was installed, both my KLR's have been rock steady.....even with full knobbies and low tire pressures (20 - 22PSI). Lastly, as others have mentioned, the rider also plays a part; keep a relaxed light grip on the bars and don't tighten up. Changing your position (move forward/lean forward) can help too.