There are two steps I would take to troubleshoot this. First if you haven't already, check to see if this is a known problem, Google it a bit.
After that if your on your own "oldturkey03" has you on the right track. With the car running, the voltage at the battery should be 14VDC or a bit more, this is most easily checked with a multimeter (if you don't have, or know someone with a voltage meter, try going to your local autoparts store, they may have one to test it for you).
If you do have 14VDC while running (12VDC when car is off), then check your battery cables for corrosion, are they tight enough? Also follow the cables from the battery to whatever they connect to next, the negative (Black) cable will probably bolt to the engine, with several other wires bolted to the same place, make sure it's tight.
From there you can start looking at all the junction points the Positive cable makes, all the way back to the fuse block. This will include alternator and starter connections.
The Negative cables need to be checked as well, after the battery they can be referred to as "Ground" cables.
A quick way to check if your alternator is working is to start the car and simply remove the Positive cable, the car should run with just the alternator. If not, check connections/corrosion then take it in to be tested.
Tracking down electrical problems can be time consuming but I hope this helps!
Jason
http://www.motortrend.com/cars/2002/hond...