Working Class identity was largely centered in urban areas, among industrial workers and slum dwellers. Unions were active here, and they, too, often employed class warfare rhetoric. But white union members often saw African-Americans as a threat to their jobs and rather than acting in solidarity with them, actively sought to exclude them from membership. Communist propaganda found a fertile field in Harlem, for instance, but not so much among white New Yorkers. Many white ethnic laborers were also Catholic, and the Catholic Church was opposed to Communism.
Real enthusiasm for Communism was found in some disaffected Middle Class people. Some, no doubt, were attracted to it out of conviction. For others, it was something that spoke to their own alienation. Years later, in the 1960s and 70s, affluent white young people often affected a working class identity in order to be more "authentic."
So the Communist Party never really caught on in America. Socialism of a limited sort was attractive to many, but class warfare rhetoric failed to hold people's attention. In an overwhelmingly Middle Class country, where "making it" meant assuming a Middle Class identity along with Middle Class income, there wasn't a lot of future for hard-core agitation.
Envy and Pity were often used in public debate, though. Pity is the more presentable emotion, and it has often been used to motivate the public to "help the poor." From Johnson's War on Poverty to the Urban Renewal movement to the debates over what to do about those without health insurance, politicians and social workers have kept the need of Lower Class America in the consciousness of the rest of the country. One could seriously debate the results of many of these programs, but the impulse to assist those in need is a good one. Meanwhile, pity moves legislation, but envy produces votes. Conflictual rhetoric tends to sharpen when politicians are speaking to the poor, rather than about them to somebody else.
The Democrats often employ class warfare rhetoric. This is partly a matter of conviction and partly a matter of convenience. Socialism and Progressivism are part of the party's tool kit; they're what they have to offer to fix America. They mean what they say on such matters sincerely. On the other hand, whipping up class anger is also a stock Democratic response to the fear of losing an election.
The Republicans remain staunchly Middle Class. Richard Nixon's "Checkers" speech, which he gave to salvage his place on the ticket with Eisenhower, was in response to charges that he had enriched himself in public service. His comment that his wife Pat didn't have a fur coat, but just an honest Republican cloth coat has been much derided. But in laying out his assets before the public, he was establishing his Middle Class bona fides, and it worked: the charges went away, and Ike kept him on the ticket.
Republicans have difficulty in talking about the poor, because all of their policy talk is aimed at the Middle Class. No doubt Republicans are sincere in their desire to assist the poor, but recommending tax credits for 401k plans to people who don't pay much taxes and don't have investments garners a lot of blank stares. Democrats are more comfortable with the rhetoric of division, even as they employ the rhetoric of unity. This comes straight out of Conflict Theory sociology.
Meanwhile, there are, of course, wealthy Republicans and wealthy Democrats, who act in ways typical of the wealthy. In public life, however, wealthy Republicans tend to present themselves as champions of the Middle Class, while wealthy Democrats present themselves as champions of the Lower Class. Upper Class behavior is fine, in other words, unless you want the support of other people, in which case acting entitled is a liability.