With the shore water hose hooked up and turned on, we have pressurized fresh water at the slip. But quietly, there was an unseen drip from behind the water regulator - you know, the thing that you screw your shore water hose to. It looks like this:

Behind this regulator, the fresh water inlet line is attached, with a simple threaded collar. You can see the threaded portion where the collar is tightened on, in this photo:

Just remove the four screws holding the regulator in place. Lift it up gently and check the fitting behind for tightness.
The threaded collar on my line was simply loose, probably from the last ten years of vibration, and I believe the previous owner was unaware of this issue. Why? The area below the regulator had water stains, and it was promoting rust on some metal parts below, which I've touched up with WD-40 oil. A simple twist of the collar to ensure tightness was all that was needed. No more leaks!
Be nice to your bilge. Have a peek down there.