I have connected an external drive to my Airport Extreme, but it doesn't show up in Finder, and it doesn't show up in the list of drives when I connect to the Airport Utility.
The same drive worked previously, but I have since reformatted it. The drive works fine when connected directly to my Mac via USB cable.
Update
I don't recall how the drive was previously formatted, except that I had both a Mac and Windows partition, and the Airport Extreme recognised both of them. Now I've formatted it as a single partition, with format: Mac OS Extended (Case-sensitive, Journaled, Encrypted)
Best Answer
I had the same problem with my Western Digital USB drive. I re-formatted it as Mac OS Extended (Journaled, not case-sensitive, not encrypted) and it worked.