If you connect it to the Mac with iTunes NOT running, and run Disk Utility, does it appear in the sidebar as a "disk"? Also, you can run System Profiler or System Information (depending Mac OS X version). It is in Applications/Utilities/, but you can also run it from the About This Mac window. Does the shuffle appear as a device under USB?
The red light when you move the power switch to ON means very low charge in battery. That probably means the battery is not getting charged when connected. In terms of hardware, that could mean a bad USB port on Mac, a bad USB cable, or a bad connection on the shuffle. In any case, power is not getting to the battery. It might be a bad battery, but if that was the only problem, I think the shuffle would still appear in iTunes when connected. That's why asked if it appeared in Disk Utility or System Profiler/Information; if it does, the connection is working (even though iTunes does not see it for some reason).
You can make sure the USB port is working by using something else that is similar (like a USB flash drive) on the same port. If you happen to have another USB cable, you also try to rule out a bad cable.