Find out how Becky can sell her booze for such low prices

Find out how Becky can sell her booze for such low prices is an unmarked quest in Fallout 2.

Walkthrough
Go to Becky's bar, ask Becky about a drink. Go to The Hole and ask Frankie about his booze. Tell Frankie that Becky's whiskey is much cheaper and offer to find out why.

Gain entrance to the Becky's bar 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.

If you don't have the patience to wait for the guard to move, you can ask Becky why her drinks are so cheap. She will tell you it's a "trade secret" and doesn't know you well enough to tell you, offering you a drink instead. Keep taking the drink from her (may take 4-5 times) but eventually she'll simply tell you she has a still.

Return to Frankie to collect the reward and access to the next quest: Sabotage Becky's still.

Bugs
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.