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.
Frankie can be convinced (with Speech around 40) to buy his liquor from Becky even after receiving the reward for destroying the still, this will net both experience rewards for a total of 1600 XP at the cost of 20 Karma.
In the unpatched game, Becky would accuse the Chosen One of destroying her still and refuse dialogue even if Frankie was convinced to buy his whiskey from her. [verified]