dim display problem/dimmer wire?

No, it sounds like the h/u is faulty. The dimmer wire from the car is activated when the lights are turned on, so if you have a dimmer wire on the deck, it dims with the lights switched on, assuming it is night.

So, I'd probably guess the contacts between the detachable face and the unit are bad. Unless that unit has the option to dim it on and off, as some units give the option.

Is it a new or used unit? 

Through experience, I have seen numerous Jensen and Linear Power units (same build house) come through with the exact same problem you are describing.

its possible but very very unlikely. you can try putting new led's in the face but that requires soldering/desoldering. the easiest route would be to replace that head unit with another one. if you want to be totally sure that its the head unit get a multimeter and check the connections and see how much voltage you are getting to the h/u. if you see bat voltage at the plug for the h/u then its more than likely the h/u.

I have a '92 Honda Civic DX.

After installing my JVC stereo, I had the same wire left over.

However, whenever I turn the headlights on, the stereo dims accordingly. So (not being an expert) I assume that this dimming function is provided by the illumination wire. At least on our cars.