B737 Captain reported a failure of the #2 Generator Drive; followed by an APU auto-shutdown due to a fault. Flight continued to destination.

Narrative

In cruise around ZZZ VOR; we had the #2 Drive light illuminate. We ran the Electric Drive QRH; which instructed us to disconnect the #2 Drive and start the APU and connect it to the affected side. About 10-15 minutes later the APU had a fault and shut down. We re-ran the drive QRH; which instructed us to land at the nearest suitable airport due the single power source condition. After that; we ran the APU fault QRH. At this time; we were about 140 NM from ZZZ; and we determined that was our nearest suitable because we were still in cruise at FL380. We [requested priority handling] and requested direct to ZZZ. ATC gave us direct; and we relayed our Souls on Board and fuel status. I sent a message to Dispatch about our status and that we were continuing to ZZZ. I then informed the Flight Attendants of our status and had them clean up early. I didn't inform the Passengers about our #1 Generator status after discussing it with my FO and Jumpseater. We were all in agreement that we shouldn't cause any extra stress for the Passengers. The weather was VFR in ZZZ; and we were also very busy coordinating with ATC. The rest of the flight was uneventful.