It depends on the context of the class I think. If there is some thematic, in-world reason for why a particular class exists, (eg. The wood elves of this forest are able to be Druids, because they've lived in the forest for centuries and have formed a bond with the spirits there), then no, I don't think any other races should have access to the class.
However since most games don't make it that specific, and just say "Some people are able to be X just because random reasons" then sure, any race should be able to be any class.
I personally prefer to have some level of specialization to things though, because in most fantasy books, there is usually a strong distinction between the races and what they can do. Many book series have it where only one race has some special ability/power, that would basically be a class to itself, so there is precedent for it, but it seems most publishers don't want to deal with the "but I want to be a *insert race/class combination* noooooow!" complaints, so they just leave it all open.