USB port on external hard drive broke. How can I access it?

I have a 2 TB Western Digital My Book 3.0. The USB Port on my hard drive broke, so I can no longer access it on my laptop or desktop via USB. Is there any other way for me to access my data? Would I have to open it up?

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.

....or purchase another enclosure of the correct size 2.5inch or 3.5inch

How would I connect it directly to my motherboard? Would I need to buy some sort of wire for that?

One of these works pretty good.

http://www.newegg.com/Product/Product.aspx?Item=N82E16812232002

Else assuming it's a sata drive you'll need a sata cable to attach it to your motherboard.