Just to be clear, if you drag down the notification bar after connecting your phone to your computer you don't see a USB connected message? I would power off/power on and try again. If it still won't connect try installing the USB drivers Motorola provides on their Droid Product Support page.
Thanks for the suggestions.
Yes, when I pull down the notification tab I don't see any notification about the SD card or the USB connection. So I tried rebooting the phone, but that didn't work. I also installed the Motorola Mobile Phone USB driver and rebooted my PC. However, unfortunately it still doesn't work.
Do you have any other suggestions? I really appreciate your help.
Further information on this problem -- it's the same one as found here. But I tried the solution suggested there and it didn't work (resetting the phone by turning it off and removing the battery, then reinserting the battery and turning the phone back on.)
I'm using the USB port that's directly on my computer, where I usually plug in my external hard drive, which works fine. But it's not recognized by the Droid, although the Droid does charge when plugged into it.