Republicans are right. The scourge of socialism is already upon us. They're just wrong about which party is to blame.
Technically, "socialism" refers to a system in which the government controls the means of production. In popular parlance, however, "socialist" has instead become a more generic right-wing slur. To the extent the descriptor signals any substance whatsoever, it's about government handouts, picking winners and losers, redistribution of wealth, or something to that effect.
And yes, some Democrats have proposed some pie-in-the-sky ideas (such as free college) that meet these vague Big Government principles.