If a drive doesn't list it's contents but does on a power reset, my guess is that it's 50-50 enclosure or drive with perhaps OS software playing a role. A partial listing would change my guess more towards the drive and probably not the OS.
There are various tests you can do (some of which I suggested earlier) but these might not be practical for you.
1) Try the drive in a different enclosure or computer (for a 3.5-inch drive the computer would have to be an older iMac or a Linux/Windows computer that can read the Mac OS file system).
2) Try a different drive in the enclosure.
3) Look at Disk Utility and see if it reports a S.M.A.R.T. status for the drive. Just about all USB enclosures will NOT report the S.M.A.R.T. status. Usually the drive has to be connected via SATA or possibly Thunderbolt to report the status - again if you have a Linux/Windows PC, you may be able to do this.
4) You can get software that will test the drive. The diagnostics which comes with the Mac will not test USB drives. I don't recall seeing a recommendation for free disk-testing software here on macrumors. I use Scannerz and I've seen a few others here on macrumors use it. Some download sites list some sort of free download - a lot of people have warned about downloading software from these sites. Otherwise, it costs $30-$40 (for a downloadable version). I bought it because I have more than a few drives - a lot of them used for backups that I test periodically.
http://scsc-online.com/Scannerz.html