Unless self-assembled, USB externals usually still incorporate status LEDs. When the device is on, irregardless of the connection, do you see a flashing status indicator showing that disk reads and writes are taking place? Some drives, without sturdier covers, will allow you to simply touch the case and feel if this activity is on-going (you could feel platters moving).
If its a working drive, you can try and swap out the USB cable. Sometimes the cable can be the source of an intermittent connection problem. An external USB drive will usually incorporate with it a controller that is separate from the drive itself. This allows for the connection of a drive that would usually use SATA to convert to USB. If this part starts to fail, you will see a recognition problem as well. What's important to highlight there is that the drive, were it to be connected directly inside the computer, could well be fine and good, but the external function that makes it USB is the part that could be failing.
You should check the LEDs, and see if the drive is alive. You should try switching the USB cable if you can, and if you have access to a second computer, you should give that a go and see if the problem persists.
If you still have the problem, one option is to completely remove the drive from the enclosure and directly connect it to a SATA port by removing the case cover on your computer. If the drive is detected and works without flaw, you know then that it is the enclosure or the cable: not the drive. This may read as being complicated, but it is, unfortunately, the way these problems are tested, generally.
If you could confirm that the drive works fine when its not in the enclosure, you could then buy a new USB enclosure for the drive, or go so far as to buy an eSATA card that connects through the back of your computer, a eSATA enclosure, and a eSATA cable. This will improve the speed of the drive significantly, making it function as if the drive is installed inside the computer itself.
My point is that it may be either the cable or the drive enclosure itself that is creating a problem for you. If you've had the drive for a long time, and have the time to experiment, this could very well solve your problem, and so therefore you may not have to buy a new drive. If the drive is under warranty, or if you don't feel comfortable doing it, you definitely don't want to start taking things apart. But one thing you can do for sure is to check that USB cable - sometimes the problem comes down to that.
Chances are the drive itself is not bad at all, but all of the components surrounding it, which make it USB compatible, are suspect. Within that case you are dealing with either a 3.5" inch regular 7200 RPM drive or a laptop sized 2.5" drive. Sometimes manufacturers will go cheap and still give you a 5200RPM drive that's 3.5" because they know you'll never reach the rotation speed using USB2.
