Easy fix. Get behind the panel (remove remote if req'd). Find individual wires not associated with stock remote cable. Disconnect red wire from current source. Put it on another source with a switch on it, or, add another switch from West Marine (single pole, single throw). This is a low amp wire and light. It isn't critical where you hook it up, as long as it isn't hot to the battery. Whoever put it there did it because it was probably hooked to some odd switch like nav lights (like on my 340).
John is correct. That was an issue I had when we installed a new remote. In my case the illumination lead was on hot and it somehow would allow the light to come back on when battery switch was off. We had installed per Clarion instructions. MM