In response to an email I just received through my Web Form Mail ...

No, the drive isn't bad. Rather, the floppy drive ribbon cable is backwards on the floppy disk drive.

Shut down your PC. Open the side panel. Remove the ribbon connector that's attached to the floppy disk drive and plug it back in, up-side-down from what it was. There is probably a small "channel guide" in the center, on one side of the ribbon's end. Use it as a reference.

Also, usually on floppy disk drives, the ribbon will have one red line on the very edge. That red line part of the ribbon should be the closest (or pointing to) the 4-pin power connector that attaches to the floppy drive. (NOTE: there are exceptions to this, depending upon the drive manufacture).

After everything is reconnected, power up the PC. The light on the floppy disk drive should now NOT be always on. After you boot to your desktop, insert a blank floppy disk and try to format it.

Additional input and installation instructions from Forum Members is welcome. Thanks.

I had that problem once.
I installed a floppy drive in an HP Pavilion a1101n. When I booted the computer, the floppy drive's green light was constantly on. I thought that it could be the ribbon cable because for me when I installed the drive, it did not have any notches like on a typical cd drive so I plugged it in backwards. I did the red to red method and it worked.

However I don't have any floppy drives in any of my computers because they hold so little data.
Store bought computers don't even come with floppy drives any more.
I wonder when they'll stop making floppies all together?