Most common problems center around chain and sprockets. As mentioned above, starting off with NEW everything is a good idea and would probably take you at least 15K miles into your trip.
Tires are the other issue. But you CAN find tires in Guatemala and Colombia for sure. Rear tires wear out, fronts last double rear.
Rear brake pads may wear quickly but not so much fronts when touring. As mentioned, riding off road in mud pads wear quick. I'd pack one set of each for 20K plus mile long trip.
Tubes: I carry 3. One on front fender (21") two 17" in luggage. Hard to find quality HD tubes, but tubes are for sale, mostly Butyl Chinese ones except in a few big cities.
Learn to use Llantero shops. Let them patch your tubes. They do a good job. Never toss out an old tube until its truly dead and you're sure you can find another. Remember, a 17" or 18" tube works fine on KLR. A 21" will work in rear tire too for short distance. (1000 miles)
You can't cover every eventuality, but doing really good, comprehensive service before departure will avoid a lot of common problems. A good long shake down ride is also recommended. (2000 miles or so)
1. 2003 is an OLD BIKE. So Check all bearings, re-grease or replace (head, swing arm, wheel, linkage)
2. New DID X Ring Chain, carry two spare front sprockets, swap them out every 8K miles. This will extend chain life by a lot.
3. 3 tubes, quality natural rubber ones. (1 21", 2 17 or 18")
4. A tool kit you've tested out pre-trip in your garage (bead breaker?)
5. couple spare jets for altitude, spare pilot jet.
6. Carry any part that commonly fails on the KLR. (Stator? Pick up coil?)
7. Old bike, so make sure cooling system is perfect, hoses OK, fan OK, water pump OK.
8. If your KLR uses oil, keep close eye on level. DAILY.
9. New bolts done for sub frame/rack?
10. NEW high quality battery. (I like Deka)
11. Make sure valve gear has enough life left and that compression is GOOD.
The good news about the KLR down South is that so many guys have made the trip, you may find OEM spares from crashed or junked bikes along the road. We hear of these once in a while. Good source for parts perhaps?
Establish an account with a really good Kawasaki dealer at home so parts can be shipped to you immediately if you need them. Waiting around down South can be expensive.
!Que le via bien!