Well.. The charging circuit has always been the weak spot on aircooled GSX's.. The stock reg/rec is the culprit, but even aftermarket Electrex regulators can break (I managed to break 2 units..) The voltage regulation is poor, and units close to breaking down are easily identified by the indicator flashing rate going up/down with the engine revs..
First actually measure the actual charging voltage both at the battery as well as the AC voltage from the alternator (as described in the service manual) to determine if the fault lies with the reg/rec or the alternator is fried as well..
Replacing the Electrex for a more modern mosfet-type regulator (like the Shindengen FH010/FH012/FH020) is a very good idea, as it is very reliable and delivers a constant voltage unlike the older type regulators..