Originally Posted by Bluto
It looks like some real boaters are chiming in, so I will defer to their experience.
It would seem like the shore power wiring in some marinas can cause weird problems. It doesn't take much 'noise' to dimly light a small LED.
Bluto is most probably correct but a check is worth while.
disconnect from shore power
Close (turn on) all your circuit breakers
Set multimeter to ohms/resistance
insert one probe into the neutral slot in an outlet
and one probe into the ground slot
There should be no continuity between ground and neutral
a few intricacies ...
If you have an inverter make sure it is turned off
If you have a generator make sure its off and its breaker is open
If you test at a GFCI protected outlet you are permitted no less than 25megohms resistance (ABYC).
If you see continuity (less than 25megohms) between neutral and ground the problem is on your boat and it is significant.