I've just installed SC1+2 and gave SC1 a try.

The default controls seem to be WAD to move, and TAB (special), LEFT SHIFT (fire primary weapon).

This is a very awkward control layout. I'd like to move the Special and Fire to other keys, but I don't see how that's possible. I tried running the KEYS.EXE, but it gives an error about OS version.

Start the game and press Crtl+F1
That opens DOSBox' key mapper, where you can remap the keys.

Thanks. That's definitely getting me closer to the goal. But the window that opens isn't very self-explanatory. How do I select the "Fire Primary Weapon," "Fire Secondary Weapon," "Accelerate," "Rotate Clockwise," and "Rotate Counter-Clockwise" events? I don't see anything that looks like these events that I can select and then assign to a key.

Yes, it's a pretty old-school interface. You don't assign game events to keys, but the original key for a certain game event to a new key.

Here's the section of DOSBox' manual explaining how to use it (still a bit cryptic ;-)
http://www.dosbox.com/DOSBoxManual.html#KeyMapper
(With "events" they mean input events, like keyboard keys, mouse clicks and joystick movement)

You have to run keys.exe through dosbox, this allows you to remap the keys the traditional way.

I believe the first post in this thread explains how (it's not as easy as it should be!)
this thread

Those instructions may be for SC2, in which case just point to the starcon1 config files instead.