The issue isn't offense-taking, it's the piling on. I'm 100% good with people saying, "Hey, what you're calling an innocent joke is actually a really shitty, ugly thing to say to someone." That's a valid complaint whether it's directed at a professional comedian or a friend at a coffee shop. What the internet does is make it very easy for 100,000 other people to come screaming onto the scene purely because they see a chance to kick somebody. They're just there for the conflict, because it's fun to put an asshole "in their place." You want to know how to tell the difference?
It's pretty easy: When the subject apologizes OR if it turns out that the subject never did the thing they were accused of, do the critics admit they were wrong? Or do they just keep right on piling on, using the apology/evidence of innocence as fuel for the fire? ("Oh OF COURSE he denies saying it!!!").
That's where it ruins things for everybody, because when a valid complaint DOES come along, it's easy to dismiss it as, "Oh, it's those PC thugs piling on again." And that bad, because there is tons of shit that's worth calling out. Humanity is still full of just awful attitudes toward the most powerless groups among us. But all of this is why on Cracked we actually try really hard to walk through the reasoning behind our criticism, not just casually calling someone a racist but explaining why seemingly innocuous things can be toxic.