I don't think a problem with the battery could cause this sort of problem. A dead battery usually just causes lost saves.
I read somewhere that some people had problems plugging the G6 into a USB hub, and had to plug straight into the PC to get it to work. Assuming you have already tried that, you probably just have a defective unit on your hands.
Did you have this problem the first time you tried it, or did it work for a while and then stop working? Does it work in the DS?