"The body pack transmitter for my wireless system I use on stage has a long cable (antenna?) on the bottom. Is it okay if I wind this cable up so that it doesn't 'hang'?"

One of the smallest and simple antennas that is consistent with reasonable transmitter output is an antenna that is physically (and electrically) one quarter as long as the wavelength of the radio wave frequency being transmitted. This is called a "1/4 wave" antenna. The "long cable" you're referring to is most likely this type of antenna. It takes different forms depending on the type of transmitter being used. For some body pack transmitters, the antenna is a trailing wire cut to an appropriate length. In other designs the cable that attaches the microphone to the transmitter may also be used as the antenna. In either case, the antenna should be allowed to extend to its proper length for maximum efficiency. The effective bandwidth of this antenna type is great enough that only about three different lengths are required to cover the high-band VHF range. Having said that, whether or not it's "okay" for you to wind or coil it up should probably be based on whether or not you're getting the performance out of the transmitter that you need. Try it both ways (coiled and extended) and make the call. As with all cable management, if you choose to coil it make sure you're not winding it so tight that you're causing damage to the cable or its connections.