You can write practically non-allocating code in Julia, but there are no facilities for ensuring this in a verifiable way.
I think that using Julia for practical crypto (with a moderately well-equipped and motivated adversary) is dubious. And while you can tweak the language in various ways, the consensus is that good crypto solutions are built from the ground up with the appropriate design decisions.
Julia targets an entirely different (one could say opposite) niche: convenient fast prototyping for generic, composable scientific programming code.