The short answer is that it wasn't really enforced, not that attempts weren't made. It became very obvious, very quickly after the Volstead Act went into effect that there was simply no way the government resources allocated to stopping the illegal flow of alcohol had any chance of actually doing so.
The Department of the Treasury was in charge of enforcing prohibition, so the people in the streets doing the dirty work of seizing and destroying alcohol networks were federal agents of the Bureau of Prohibition. Often referred to as "revenuers" or "prohis" (short for prohibition agents), there were only a few thousand of them in charge of patrolling the open border with Canada and making arrests in the major cities. The Coast Guard helped where it wanted to with smuggling by sea, but it often was bribed to look the other way.
The courts were soon overflowing with liquor citations from those who were simply drunk in a speakeasy, to those who ran them, to those restaurant owners who wanted to offer their patrons something not on the menu. There was no way the courts could keep up, or sentence them to more than a fine usually because there was nowhere near enough jail space.
By the late 1920s, enforcement had been so spectacularly unsuccessful that even temperance movement members had surrendered to the fact that legalization would mean less crime and violence.