Go to Becky's bar, on the other side of town. Wait around 2 minutes for the guard near the door to move away and talk to the Craps dealer. Unlock the door by any means and go down the stairs, use a crowbar on the still. Sneak back out and return to Frankie for the reward. After destroying the still, Becky refuses conversation with the Chosen One.
Alternatively, Frankie can be persuaded (Speech check) to buy his inventory from Becky, which awards 900 XP and +70 Karma.