Wayne,
This is most likely an issue with your programming cable and/or driver.
I your programming cable has an unauthorized copy of a Prolific chip (most do), you need to use a specific older Windows driver to be compatible with the chip. An incompatible driver as been sometimes known to cause the radio to key up.
XP = v2.0.2.1
Vista/7/8 = v3.2.0.0
See the miklor.com website for a link to a compatible driver and instructions for how to install it.
http://www.miklor.com/COM/UV_Drivers.php
Also it is common to not get the plug of the programming cable fully plugged into the socket of the UV-5R family of radios. Manufacturing tolerances can have the jacks not centered well with the holes in the case which causes alignment issues. Also the shell of the plug can contact the case making it difficult to fully engage the plug into the socket. A plug that is not fully seated in the socket can cause the radio to transmit.
See the miklor.com website for suggestions on how to get the plug to be fully seated.
http://www.miklor.com/HT/HT-ErrorMess.php#error1
If neither of the above solve your issue, you can program at least of the radio's channels as RX only (TX inhibited) u