I believe it absolutely has coarsened those things. The ability to say something nasty to someone and a/ not be around for the aftermath of hurt it causes and b/ be in no risk of a smack in the mouth have allowed people to lay off their inner editor. However, while I bemoan that, I don't think that means I have to advocate legislating about it, does it?
The concept of 'punching up' does amuse me - not least because of its inherently bigoted assumption of superiority on the part of those being punched. Thus, to the white, middle class liberal mentality, a wealthy Indian or African person may slag off a poor white English person because they are 'punching up', which actually only serves to endorse and reinforce ingrained ideas of racial hierarchy.