The black supplies power to the horn, the light green is the ground wire that goes to the push button. When you push the horn button, you're completing the ground circuit, and that's likely where the problem lies - in a corroded push button.
However, to verify the horn itself, hook a test wire up to the terminal where the ligh green wire connects and touch it to a good ground, like a clean unpainted spot on the motor. With the key on, that should beep the horn. Then you know the problem is in the button.
Or, you may have both - a corroded button AND a horn that doesn't work. Many times the horns will corrode up inside and you can free 'em up by lightly tapping the sides with a screwdriver handle and/or working the litte tone adjuster screw on the back in and out a few times and/or energizing them to free 'em up and get 'em working.
Good luck with it.
BTW, have you checked out the HondaTwins site? It's a bit more 'twins' focused than the 'Other Bikes' section of this 'fours' site.
http://www.hondatwins.net/forum/
