I thought most mini's have a Firewire connection, either 400 or 800. However, if yours doesn't, use USB.
If you're backing up, for archiving purposes, fast speed is of less importance than if you're offloading your hard disk and will be accessing the external backup as the sole source of your photographs.
I have a mac mini. I need to back up all the photographs. There seems to be no port for firewire. How do I get a fast connection?
These links show the location of the FireWire port on each of the different models.
I hope this helps.