How old is the battery? If it's not a sealed unit, make sure it's got an ample supply of distilled water.
Often times, parasitic losses can drain batteries on bikes that sit or do not get started frequently. Same as some automobiles. However, in this case, there could be something else in play causing these issues.
Is the bike fuel-injected? I'll assume so (my 2005 c50 is a cruiser that just recently inherited its fuel injection & management system from the GSXR), as you're having both electrical and what seems like what might be fuel management issues. That said, the two might be unrelated--you might have a bad/old battery and you might have a tank of poor-quality fuel.
Do you use Sta-Bil or some other treatment in the gas tank when you let the bike sit for extended periods? If not, another thing you can try is to give the battery a good charge, fuel the bike with fresh gasoline, and see what comes of it.
Good luck.
