Transistor biasing. Haven't done that in a decade or two. Gotta go pull that out of the backup tape in the basement archives.
Lessee...an NPN transistor will conduct when the base/emitter junction is forward biased, meaning the base is at a higher voltage than the emitter. You want it on when the connection is open? Put your load between Vcc and the collector. Then maybe you tie the base to Vcc through a couple 10K ohms. That will keep it on until you pull the base low to shut off. Would that work for you?