A PO fixed this on our boat by adding a wooden profile with a drip lip to the bottom of the door. He even chose a matching sapele wood, but sadly left it untreated! I've since cleaned it up, rounded it over a bit and varnished it. Looks like it was meant to be there now.
Ultimately we protect the door with a shower curtain though (and with showering on the transom in summer).
Can snap a picture of the drip profile if needed.