So wait, you're driving in and you want to park on SI? Forget it man, that makes no sense. Go to NYC Parking - Find. Compare. Save. | BestParking and find a garage that meets your price requirements, or go park in Queens for free. I wouldn't worry about bad neighborhoods, NYC is pretty safe but some of the worst neighborhoods in NYC are on the North shore of Staten Island around the projects, or deep into Brooklyn / up in the Bronx where you're not going to be. Being in Lower Manhattan you can take a round trip any time on the Ferry for free and not have to worry about hauling luggage.
Appreciate the tips! It sounds like SI is a bad idea for anything other than a round trip ferry to see the statue. Taking a brief look the parking will add up over 5 days. Is free overnight street parking in Queens fairly easy to find then?