The DSM identifies Alcohol Use Disorder as a mental illness, with varying grades of severity.  Alcohol abuse is clearly tied to violence - NIH ties estimates that 80% of homicides have an alcohol factor.  Therefore, AUD can rightfully be considered to be a mental illness with strong tendencies to violence.

The NRA and other gun supporters have advocated that the mentally ill with disorders associated with violence should be isolated from guns.  Since severe AUD qualifies as one of them, anyone diagnosed with severe AUD should be prevented from purchasing or possessing guns until they've undergone effective treatment.  

Those with AUD are much more likely to be convicted of DUIs.  Anyone convicted of a DUI should be required to undergo evaluation based on DSM standards to determine their AUD status.

Back to my original statement - regulate people, not guns.  And focus on those that break laws or pose a clear threat of violent crime, like alcoholics.