The fact that a USB external drive is recognized on the USB bus does not prove that the drive itself it OK. It only proves the USB bus in your computer can communicate at least partially with the controller board in the external unit.
One of the better ways to start diagnosing a troubled external drive is to remove the HDD unit itself form the external case and then mount / connect it internally in a desktop machine where you can test it. Two things you must watch out for, though. Doing this will void the warranty on the external unit, so if it's still under warranty you should contact the manufacturer instead. Secondly, what you will need to connect it inside a desktop unit depends on what type of HDD unit is inside the case. If it is a small portable PC type of HDD unit its connections and size may require an adapter to get it set up in a desktop machine. And of course, you'd need access to a desktop machine!
If you can move the unit this way it eliminates extra hardware that may be malfunctioning. If that is all that's wrong - the external case's controller is faulty - the HDD itself may function perfectly when inside the desktop. But if that does not happen you probably can get HDD diagnostic utility tools by free download from the manufacturer of the HDD unit (not the maker of the external drive system). You can use these to test the HDD and determine whether it can be repaired easily.
Oh, there's one other type of malfunction that can be fixed easily. On some occasions the connections inside the external drive case to the HDD unit are poor, and simply disconnecting the drive and re-connecting it a few times might clean that up so it works again. That's if you are really lucky!