YMMV but I don't think caulking is a very good fix for the tops of boards
that are attached to something else.

Better, IMO, is pressure treated lumber. Barring that, bevel the top edge
of the horizontal board; even better is to bevel and bed. One could use
caulking to bed...one could also use plumber's putty. Perhaps the best
bedding material is "boatyard bedding compound" which is essentially
plumber's putty with a wood preservative.
dadiOH

Only cheap solution I could come up with was to fill the top edges of
the boards to make the water run off. I believe the lumber is pressure
treated but I'm not positive. Thought about plumber's putty and might
use that instead. I'm certainly open to suggestions.