We have a proof that AC is independent of the other axioms of ZF. But that's not a good enough criterion for deciding whether or not to use an axiom. We choose axioms because because we want to use them to reason reliably. We can use ZF to reason about mundane things like strings of symbols encoded with Godel numbering. But Godel's theorem shows that we can find axioms independent of ZF, which when added to ZF, give incorrect results when applied to strings of symbols. I think some people worry that AC may allow us to reason incorrectly in some domains, even if it is consistent with ZF. (Of course we don't know ZF itself is consistent, or allows us to reason correctly, either, but the fewer axioms the better.)

We have a proof that AC is independent of the other axioms of ZF. But that's not a good enough criterion for deciding whether or not to use an axiom. We choose axioms because because we want to use them to reason reliably. We can use ZF to reason about mundane things like strings of symbols encoded with Godel numbering. But Godel's Incompleteness Theorem shows that we can find axioms independent of ZF, which when added to ZF, give incorrect results when applied to strings of symbols. I think some people worry that AC may allow us to reason incorrectly in some domains, even if it is consistent with ZF. (Of course we don't know ZF itself is consistent, or allows us to reason correctly, either, but the fewer axioms the better.)