First, I'd try a replacement USB cable and then I'd try plugging into another computer. Just to rule out any simple issues.
If it's a 3.5" hard drive you can try removing it from the enclosure to see if it is just a bad USB - SATA bridge board. If it's a 2.5" model such as the Canvio portable ones, then you'll most likely need professional recovery as they only have a USB port which is built right on the PCB.