Better POD would be to get a Democratic administration in power when the Great Depression hits, and the ITTL equivalent of the New Deal is done by a Republican administration.
However, the Donks were in really bad shape in the 1920s, even worse than the GOP in the 1930s and 1960s, and even with the intertia of the two party system, a southern dominated, Conservative Democratic party is probably not viable if it turns out its the GOP that does the New Deal and wins World War 2. This actually might be an interesting alternative timeline. ITTL, the eventual opposition to the GOP grows out of a third party candidacy of either Eisenhower or Taft when the isolationist or internationalist wing of the party loses the debate about how engage to be in the world post World War 2. And that means the American system finally includes a major minor party as the Democrats survive as a sort of regional party.
To do this, you almost have to have a GOP administration take the USA into World War I, because that was probably the primary reason for the Donk colapse in the 1920s, and then the backlash gives you a Democratic administration in time for the Depression. But this all has enormous amounts of butterflies, not leas the effect on Prohibition.