i am also having problems with my horn. the horn relay was missing when i got the car. i got a new one and plug it in and the relay energized right away. i checked the horn switch for continunty to ground and there was none until i pressed the button which is ok. what do i check to see why the relay is energized all the tims i have an orignal gm shop manual for the 1981 and have tryed to follow the troubleshooting without any luck. is it possible the anti-theft circut is causing the problem? i have tryed to locate the modual and cannot find it. it says it is located at the lock pillar support. what is the lock pillar support? if you know what module looks like,please describe it.

So....if the relay "energized" when you plugged it in, did the horn(s) blow too? If so, then there has to be a short to ground somewhere in the control side of the circuit, but the circuit from the relay to the horn(s) is good.
 If the horns did not blow, but the relay is still "on", there still has to be a short. The relay should have battery power all the time. The horn button provides the ground to "latch" the relay closed, sending the battery voltage out tot he horn(s). Gotta be a short to ground there on the control(ground) side of the relay.
Where did you test the button for ground? At the button itself, or where the harness comes out of the column, or at the relay itself?

Good luck with your horn.  I just got mine to work today.  In fact after the 5th time taking it all apart I gave up put it all back together and now it works.  So I'm not any help with the thing.  

The did blow. I checked for continunty from the horn button to the screw on the steering wheel. I pressed the button and it made continunty.When I released the button I lost continunty as it should be.I was testing with the relay removed so I would not mess up my meter.Lookin in the shop manual it shows green wire to horns,orange wire to 12 volts,and black wire to the anti-thept circuit.I know the black wire to the relay has continunty to ground all the time without the relay in place.I do not know if the black wire goes any where else.If you can help me with some other way to trouble shoot , I will be thankful.

The black wire at the relay goes to the Anti-theft module, AND to the horn button. Your problem may be in the anti-theft module, keeping the black wire grounded all the time, OR...one or more of the anti-theft switches may be grounded, which would keep the module live all the time.
You might try unplugging the module, and see if the horn button works properly then. If it does, then I would suggest checking all of your switches...the ones on the hood, and both doors, to see if one might be stuck, or the wiring touching.

The "lock pillar" is the area of the body to the rear of the doors..where the door latch posts are. You may need to remove the trim panel in that area to locate the module. I've never looked for that module, so I couldn't say for sure where it is, or even what it would look like. Perhaps someone here with an '81 might have more info...