The discoverability is dismal, but it's quite clever how Advanced Rest Client handles basic authentication. The shortcut abraham mentioned didn't work for me, but a little poking around revealed how it does it.
The first thing you need to do is add the Authorization
header:
Then, a nifty little thing pops up when you focus the value
input (note the "construct" box in the lower right):
Clicking it will bring up a box. It even does OAuth, if you want!
Tada! If you leave the value field blank when you click "construct," it will add the Basic
part to it (I assume it will also add the necessary OAuth stuff, too, but I didn't try that, as my current needs were for basic authentication), so you don't need to do anything.