It's hard to know where to draw the boundaries on this sort of thing, and we need to distinguish questions. Is the claim that the working class will themselves be offended by use of the term "vulgar"? Or is the claim that, by using the term, you are showing them disrespect (that is, showing disrespect for them), even if they are not themselves offended? Different cases will be different.
Consider the verb "gyp", meaning to defraud or swindle. As the spelling indicates, the term comes from "gypsy". I expect few people, other than gypsies, know this. So, in that sense, people who use the term are probably not trying to offend gypsies. But, if I were a gypsy, I'd find it offensive and so, by using it, one may be offending such people, even if one does not mean to do so. One would not be blameworthy for that usage, but, once informed of its consequences, one should stop using the term.
A similar case might be "Indian summer". The history of this term is closely related to the more obviously derogatory "Indian giver". Again, few people who use the term are probably aware of the etymology, but their use of the term may still be offensive. And we might add here the related case of the use of Indian "mascots": The Braves, Redskins, and so forth.
Yet another case is the common use of the term "gay", among the younger set, to mean something like "stupid". My daughter tells me that many of her peers are completely unaware of the history of this usage and don't themselves intend to insult anyone by using the term that way. Here, one might think there is some blame to be assigned, since obliviousness is not necessarily an excuse.
All of that said, I think your professors concerns are misplaced. I very much doubt that working people are going, as a group, to be offended by the use of the term "vulgar". Nor does the mere fact that a word has a certain Latin etymology imply that its use expresses something connected with that etymology.