I'd look at the batteries as a prime suspect. If they are wired together, then yes, one bad battery will pull the other one down. The alternator is only meant to keep the batteries charged when they are at full capacity, not recharge a dead battery (I know some of you stereo folks will disagree here) but I'd pull them out and take them both down and have them tested.
I don't know about boats, but batteries in vehicles in AZ seem to go away every couple years.