Those specs are not that bad. I've run Ubuntu (rather sedately, it's true) with 6ooMHz!
The memory is rather small for Ubuntu/Mint though. It all depends on whether you have a video card; if you don't, then you haven't got 384MB: more like 320. If that's the case, go into the BIOS at the boot and make sure that the memory allocated to video is the minimum it will allow.
Distro will often matter more than the GUI. For example, Dream running Gnome uses 3/4 of the memory that Fedora does. For a good performance on your machine, I'd recommend Crunchbang; it's Debian based, like Ubuntu or Mint, and will run with 400MHz and 256MB. Vector is also nice, but use the text-based installer rather than the GUI one.