Bill is right, the wire only caries the ground to the horn relay. The horn relay has power coming in and a wire going out to the horns. To activate the relay, you supply a ground to complete the circuit. The actual push button has two wire prongs that rub on the steering wheel hub which is connected to the gear box through the steering shaft. The shaft gets it ground through the bearings and steering box housing, which as you know is connected to the frame. Thats a lot of connections that need to be clean and tight. The top bushing which you see in the column is a wood/rubber/man made type of material which insulates the brass ring from the steel tube. GM designed the horn to work off ground for a safety reason. If you was to run a hot wire up the steering column tube, you risk the chance of a direct short from chaffing. By using a ground, the most that can happen is the wire shorts to ground and the horns blow, annoying yes, but still better then sparks and possible fire you could get with a positive wire.
I believe new bushings can be had if yours is worn in the center. I made one using a piece of teflon drilled and fitted with the original brass ring and new wire. Mine was well worn letting the shaft flop around. You will also find the push button has very little down ward travel so the wire prongs set pretty close to the brass contact. its possible to honk the horns when turning hard or really working the steering wheel. If the bushing is really worn, the brass contact can hit the shaft causing unwanted honking.
The wire exits the column at the bottom through a rubber grommet, be sure the rubber is in place.
To test the horns, apply direct positive voltage to the wire contacts at each horn.
If the horns work, follow the wire going to the horn relay. This should be mounted somewhere on the firewall,
http://www.chevsofthe40s.com/images_product/large/1116775A.jpg. One wire will be hot, one ground, and one to the horns. The ground will only be active when the horn button is pushed. Hot wire will be direct battery voltage, 1937 has a circuit breaker feeding the amp gauge and horns. A quick ground to the relay will let you know if its working, and volt meter or test light will tell you if it has power. If everything so far works, you are down to the horn button and wire.
Joe