'83 Toyota pickup start trouble in cold, wet weather

While bedlam and fish's suggestions could be the cause, it does sound more spark related.
Check that it's getting fuel when you pump the throttle prior to starting.
Seems electrical to me since the moisture seems to cause the issue.
Cap and rotor are cheap and if they're old, it's a good idea anyway.
Or may e first shoot the underside of the distributor cap with WD40.