The first I encountered that was in SILC, where it was used to make the key fingerprint more or less pronounceable. It's interesting that the encoding scheme never really got wider attention, although it is available in both openssh and openssl.
Similar ideas come and go, and get rediscovered - so maybe now is the time for wider acceptance.
A bit of googling provided me with a nice starting point for implementations, so babble print has certainly been recognised in some circles.[1]
That only works in English, again.
Of course that's not helpful when two people who don't speak a common language are trying to establish communication, but it probably isn't their biggest problem.
They already spent a bunch of time and effort finding these phonemes to build Esperanto, right?
Its choices are large, irregular, unclearly defined and basically Eastern Polish.
Lojban is designed to be well-defined on phonemes, but that still only works when knowing its (rather simple) pronunciation rules.
[1] http://web.archive.org/web/20090918202746/http://tothink.com...
I rather like the urbit way of encoding numbers. I can't remember the exact details but it's something like: There are 256 unique three letter words (all nonsense, but deliberately picked to be possible to pronounce). Each word is mapped to a byte and then a blob of binary data becomes a nonsense word.
So for example "tasfyn-partyv" instead of "1242B24A".