World War I was a major watershed for the parties as they realigned themselves into new poles of opposition. Woodrow Wilson's internationalist faction had taken control of the Democratic Party and ended a long Republican reign in 1912. They broke from Teddy Roosevelt's dictum of "speak softly and carry a big stick" toward measures of what today would be called "collective security". This eventually evolved into Wilson's proposal after the war for the establishment of the League of Nations (which the US would not end up joining). In response, the Republican Party became the place for people who wanted the status quo ante bellum of isolation to gather, and the isolationists became the dominant factor in the Republican Party as the war ended. This faction combined with a pro-business faction that was a backlash to Roosevelt's "trust-busting" attitude that migrated out of the Republican Party and into the Roosevelt/LaFollette Progressive Party.
In 1920, the Republicans put up a vacuous tool of isolationists and business interests named Warren Harding as their candidate for president. He was a looker as well, which the Republicans felt could help them win due to the fact that this was the first presidential election in which women could vote, and their thought was "handsome plus uninformed female population equals vote appeal". Unfortunately, they were right. By that time, Roosevelt was dead and Wilson was dying, so their influence started to wane quickly. The US refused to join the League of Nations and business interests started to become paramount in Washington, which led not only to Calvin Coolidge's famous statement "the business of America is business" but also to situations like the Teapot Dome scandal, which involved the Harding Administration selling off federal oil reserves under the table to its cronies, who bribed their way into favor. The isolationists would remain a powerful force in the Republican Party until the 1950s.
In the meantime, the Democrats became a place not only for internationalists, but also for the growing element of social democrats, who felt that Socialists and Communists were too far out of touch with what America really stood for. In order to carve out a firm footing on the left to oppose the Republicans' rightist views, the Democrats became a party that looked out for the interests of people rather than business. At this point, the modern stereotype of the two parties took shape. Unfortunately, this didn't appeal to the public during the growth of the 1920s. However, when the market crashed in 1929, the Democrats were ready. When Franklin Roosevelt was elected in 1932, he advocated an expansion of government to provide a social safety net for a poverty-stricken nation. This became the New Deal, and provided Republicans with ammunition for their cause of keeping government out of business. The Democrat-big government/Republican small-government division was now firmly entrenched.
When Europe went to war again in 1939, the Wilsonian wing of the Democrats proved they were still alive and well, and Roosevelt was a Wilsonian through and through. However, American public opinion was against becoming involved yet again in a European war, until Pearl Harbor. At that time, political matters were put aside for the duration. The Republicans, however, stayed busy. When collective security and internationalism became an issue again during the war, the Republicans divided into two camps: the old isolationists and the "it's necessary, but we really don't approve" moderate group. The moderates became the dominant faction in the Republican Party after Tom Dewey put up two good fights in 1944 and 1948 and Bob Taft was Eisenhowered out of the nomination in 1952. In the meantime, Roosevelt Democrats remained dominant in their party, and the liberal traditions handed down from them are still the dominant characteristic of the Democrats today.