Those New Reno families charge an arm and a leg for their booze. I don't know how she does it!”— Frankie
Gain entrance to the Becky's basement: pick the lock when the door guard walks off to pester the nearby croupier or destroy the door with an explosive. Come downstairs and find the still.
Return to Frankie to collect the reward and access to the next quest: Sabotage Becky's still.
Destroying the Becky's basement door might make the game unstable. Saving/reloading or exiting/re-entering the map right after destroying the door will prevent possible crashes.