Does it make any strange noises? Things like a steady click that you could set a watch to could indicate a failed drive.
First thing would be to try a different USB cable if you have one available. Trying in safe mode is worth a shot, too. Anything that lets you run a health diagnostic or chkdsk on the drive.
If you feel adventurous and don't mind voiding a warranty (if it hasn't expired), you could try removing the drive from the enclosure and connecting it directly to your computer internally to see if the USB adapter failed. If the drive works, all you'll need is a new enclosure ($20). If not, and it was still under warranty when you opened it, you'll miss out on returning it to Seagate for the almost-free refurbished lotto.