I have been experimenting with using UUIDs as database keys. I want to take up the least amount of bytes as possible, while still keeping the UUID representation human read
This is not exactly what you asked for (it isn't Base64), but worth looking at, because of added flexibility: there is a Clojure library that implements a compact 26-char URL-safe representation of UUIDs (https://github.com/tonsky/compact-uuids).
Some highlights:
These are rather nice properties. I've been using this encoding in my applications both for database keys and for user-visible identifiers, and it works very well.