Short of taking it somewhere for repair, it sounds like you're going to have to open up the case to get inside the drive. Once you've done that, you have two options:
1) If the connector is still functional you can connect it to your computer with the case removed and transfer your data, or
2) You can disconnect the bare drive from the guts in your case and connect it directly to your motherboard so that you can read the data from it.
i have the same problem with my hard drive as the previous user...the usb port is broken and fallen inside. if i were to open up the case (i presume i would have to bust it open, i dont see any screws holding it together...) what can i do to prevent damaging the info on the hard drive...are there certain parts i shouldnt touch or....? losing the files i have stored on this drive is not an option, they are extremely important and irreplacable......
As long as you don't have an issue with static electricity, merely touching the drive or the components inside the case won't cause any damage. The most important things to watch out for are to avoid any strong mechanical shocks to the case while you try to open it, and don't use a tool that might accidentally get thrust inside the case where it could hit some circuitry and damage it.
If the drive is a 2.5" Passport, then it will have the USB port on the hard drive's PCB. Your only real option then is to have someone resolder the microUSB connector for you, even if they have to attach it with wires (eg wirewrap wire).
OTOH, if the external drive is a My Book, then it will have a separate USB-SATA bridge board. This then gives you two options:
(1) If the drive is an Elements, then the data will not be encrypted, in which case you could connect the HDD directly to a SATA port on your computer's motherboard, or install it in a third party enclosure.
(2) If the drive is an Essentials, then it will incorporate 128-bit AES hardware encryption, even if you have not set a password. This will mean that your data will appear as gibberish if you use one of the options in (1). In this case your only solution, other than resoldering the connector, would be to purchase an identical product of the same capacity and swap the bridge boards.