I think washing the board is a good idea. There could be a short from 12V to ground or to 5V caused by capacitor leakage on the board. Worse, would be an internal short in one of the bypass caps on the 12V plane.

But if the short was 0 resistance, I don't think the power supply would start up. If there is a short, it is probably somewhat resistive.

I would start by disconnecting the power supply and the disk drives and then check the resistance between 12V and ground and between 12V and 5V using an ohmmeter.