I'll go along with a less-developed version of organized crime: by all accounts, prohibition essentially made it what it is and financed future expansion. The FBI? Not so sure: what we know now as the FBI came as a result of an overhaul of the scandal-riddled Bureau of Investigation (a thoroughly upright creation of TR, first headed by Charles Bonaparte, which became corrupt indeed during the Harding years) during Coolidge's tenure. Absent prohibition and assuming Harding's presidency and death played out as they IOTL (not difficult to imagine given any association is loose at most), I don't see much to change this.
I wonder if a side effect--namely, a more-or-less tacit grudging acceptance of the federal government become part of day-to-day life--came about as a function of prohibition that helped to make New Deal measures more acceptable? Granted, people would have bought nearly anything FDR said if there was any hope of relief, but still...also, one wonders if the general mentality of the public during prohibition led to relatively unbridled stock market speculation, inflating the bubble that burst in 1929? That is, without prohibition, would (ironically somewhat) a more conservative financial mentality have prevailed, yielding a lesser event--say, more of a recession--in 1929 that Hoover might conceivably have been able to handle and ride out, assuming he'd been elected?
