I would do some basics. Remove and reinstall each bulb to ensure good contacts. Maybe even clean up the contact studs on the bulbs with a little emery cloth or steel wool. Check the fuse. Check the flasher module. Mine is under my seat on my GL500. The older style ones do go bad after a while, so it may be worth it to buy a new one. If none of that works, find the electrical diagram on the wiki page and start tracing wires out, looking for shorts. You could use a multimeter and check for continuity. Electrical stuff can be hard to track down, but at least these old bikes don't have a ton of wiring. Good luck.
I would think that this could be a vibration-related problem. Check the connections around the flasher relay, especially. Also possible that the internals of the flasher are close to dead. Though I don't normally suggest replacing parts on a hunch, it's cheap and it might be worth doing.