Bad starter?

I have a 1991 Ford Explorer. When I turn the key, not much happens. No click, but I hear a brief hum. The battery is good. Is it worth changing the solenoid (which I can do myself) or is it most likely the starter?

You need to check what is causing the hum. It may be the starter motor turning but the bendix may not working and moving the gear into position to turn the ring gear of the engine. If that is the case then the starter will need to be worked on or replaced.

The hum could be unrelated to the starting system. It might be the fuel pump, for example. Your starting problem could be due to practically anything. The ignition switch might be the culprit. Grab your voltmeter and do some troubleshooting before randomly replacing parts.