Expansion on usage of the phrase
Tvtropes.org lists this line as an example of the “Screw the Rules, I Have Connections!” trope.
From that page:
A character is able to [ignore] the rules, simply because their friends or family are very influential, powerful, or wealthy people.
Source: tvtropes.org, Screw the Rules, I Have Connections!
In The Year’s Work in Lebowski Studies, this line is used as evidence of Treehorn’s money, influence and power (my interpretation of “conventionally successful”).
. . .[Treehorn] is probably the most conventionally successful male figure in the film.
Source: The Year's Work in Lebowski Studies, via Google Books
Sam Bloch of the LA Weekly provides the following insightful commentary (emphasis mine):
At one point during The Big Lebowski, our hero, the Dude, finds himself at a mysterious beach party thrown by pornographer Jackie Treehorn. Inside his modernist home — the ultimate bachelor pad — a fire blazes as the Dude takes a seat on a long, orange leather couch beneath a coffered ceiling. Treehorn, played by Ben Gazzara, prepares him a spiked White Russian at the bar. A swimming pool is separated from the angular living room by a wall of glass, the mise-en-scène daring the viewer to look more closely at a villain professing transparency. As the Malibu sheriff later informs the Dude, Treehorn draws a lot of water in this town.
The phrase appears in the context of justice (c.f.: Playback: A Novel and Personal War), with the implication that the person who draws a lot of water can do as they please. In the film, The Dude (who is called a “goldbricker” and a “bum”) is attempting to apply his own moral code in a situation where agents of legal enforcement and society at large are more concerned with someone’s material success than their moral character.
This is a microcosm of the grander struggle between the wealthy Jeffrey Lebowski and his penniless counterpart of the same name, in which the questionable moral choices of the former have resulted in damage to what little property the latter has. “The bums always lose!” quoth the big Lebowski, and, given the trials endured by The Dude in search of meager restitution, it seems that might be right.