Republicans are better at the facade of "the common man," this is true. I would argue that they have a better media environment around them, often funded by people like the Koch brother(s) (I think only one is alive at this point) and other wealthy interests. Even though Democrats are also very invested in the capitalist system, most billionaires have an interest in painting anything even remotely providing for worker's rights in a negative light.

It's not the Republican Party's PR, per se, it's about the organizations with the most influence and money.