It so often happens that colloquialisms enter the language unannounced, gaining currency in the spoken word well before they ever appear in print. Unfortunately, in the eras that came before voice recording, print is often the only record we have. I went into Google Books and found a few examples of usage, but nothing that points to origins.
Some examples from that link:
For two streets Johno kept complaining to the driver that it was a nice banjax if a fellow ... (Sean O'Faolian, A Nest of Simple Folk, 1934)
The lady smiled at him and flicked through her mental social files till she came to the Banjax dossier and looked under " Interests." (Ireland Today, 1937)
What is meant by "the Banjax dossier" is unclear, but it possibly refers to an imaginary collection of screw-ups (AmE) or cock-ups (BrE).
The earliest apparent occurrences in print seem to be from the 1930s, supporting your assertion, which suggests that the word was in common (Irish) parlance for some years before surfacing in that form. It's too bad that the Google Books result doesn't even include complete sentences, and that there is no preview available for any of the results.