Thank you, neutron_bob! Your advice was correct.
Here’s what I did in detail in case anyone else runs into this problem.
Connect player to the USB port.
Start the following program: Start menu–Administrative Tools–Computer Management
Then open the System Tools subdirectory in the left panel and select Device Manager.
Open the Portable Devices subdirectory in the right panel and the Sansa Fuze 8GB (or whatever your player is) should show up there. Double-click on it to open the Properties window. Select the Driver tab. Click on Uninstall.
Plug player back in.
Windows XP (in my case) automatically detected the new hardware device and installed the correct driver. The Sansa Updater launched itself and I was able to download new firmware.
Don’t hate the player, hate the game.