I had a similar thought a few months ago when looking at ways to encode data in HTTP cookies and came to the conclusion that base64 wasn't broken, so there was no need to fix it.
For medium-large messages I would tend to use JSON/msgpack where appropriate, so no need to further encode anything.
For small (binary) messages, other encodings (e.g. ascii85) often wasn't much smaller and didn't encode faster, so there was no measurable performance benefit. And given the extra complexity and compatibility issues it would take to use something else, it just wasn't worth it.