In the form you've presented the claims, there would be a fallacy of denying the antecedent. If free, then responsible. Not free. So, not responsible.
But I don't think philosophers typically agree with the conditional claim, which says that having free will (or doing A freely) is sufficient for moral responsibility (or being responsible for A). And we should not agree with it. After all, I might freely decide to back my car out of the driveway and in doing so run over the sleeping cat I could not be expected to have seen. If so, I do not seem to be responsible (blameworthy) for killing the cat. There might be ways to fix up the terms, but there is likely an epistemic condition (a justified belief requirement) for responsibility that goes beyond the free will (or control) condition.
However, it is more plausible to say that moral responsibility (being responsible for A) requires free will (that one did A freely, or did some earlier action freely that one should have known would lead to A). So, if I am responsible for killing the cat, I must have free will and must have exercised it in such a way that led to my cat killing.
Suppose we accept: If one is responsible, then one has free will.
Then, by modus tollens (or denying the consequent--i.e., saying we lack free will), we validly conclude that one in not responsible.
Some people suggest that it is so implausible (and/or costly) to assert that humans are never ever responsible for anything at all (e.g., that no one deserves blame for anything) that we have good reason to question any argument (or premises) that concludes that no one has free will.