If it's less than one year old, it should still be under warranty. If you are near an Apple Store, I'd make an appointment at the Genius Bar to see about getting it replaced.
There are two likely possibilities. The cable used to connect it to USB has gone bad. Or the connector on the iPod has gone bad.
Be sure to try these things, if you have not already. Use a different USB port on your computer.
Disconnect iPod shuffle and completely shut down (power off) your computer. Also, use this procedure to reset the shuffle
Turn on computer. Run iTunes. Connect iPod shuffle.