In $\sf ZF$ the two are equivalent:
- The axiom of choice.
- Every $\beth$ number is an $\aleph$ number.
You can prove this by going through the following equivalent statement,
- The power set of an ordinal can be well-ordered.
So if the axiom of choice fails, we know that there is some $\alpha$ such that $\beth_\alpha$ cannot be well-ordered. Interestingly, the least such $\alpha$ can be a limit cardinal, and in fact it can be $\omega$.
Other than that we cannot say much. We know it is consistent that $\aleph_1$ is incomparable with $\beth_1$, for example, in which case there is no $\beth$ number (other than $\beth_0$, that is) which can be well-ordered.