Yep. Tiki's got it. This has happened to me.
Your fix is to remove whatever gunk is plugging up the float needle. Some have tried reverse spraying stuff up there, but you may as well pull the carbs and clean the system. Blow the fuel hoses clean and make sure your tank filter is working. You probably either have gummed up float needle/valve or some trash that is preventing the needle from closing. Last time it happened to me it was a paint fleck from the petcock that fell into the hose when I was reattaching the tank.
MF Battery, Iridium Plugs, RR relocated, Voltmeter, 170/80/15 Kenda Kruz rear tire, DIY Samsonite hard bags, DIY shaved seat with Beaded seat pad ('cause that's how I roll) and the dreaded STATOR CHANGE.
Shining Black Bess '86 VN750, retired for parts after a fried stator and being knocked (kee-runch) in her parking spot.