Not all humor is offensive. I love jokes. I think lots of things are funny. My favorite show is Whose Line Is It Anyway (which can be used as a great example of the kind of comedy that is funny without making you look like a shithead)
Sometimes though, there are assholes who like to just say whatever they want with complete disregard for anybody else and then laugh and call it "comedy" as a way to try and justify what they said and did. It's like how a schoolyard bully might push and shove someone or maybe even go so far as to tell someone they should kill themselves or that they hope that person gets raped and then they turn around and say "I was just joking, it's their fault if they took it seriously" because they got in trouble for what they said then turn around and say the people who got upset over it are "killing comedy". Comedy is not a word you can just throw on anything you say and then it instantly becomes excusable and/or justified.
Sometimes you're just plain in the wrong for what you said, joke or not. And sometimes, you're just not funny.
I know of at least two "comedians" on YouTube who do this: Nicole Arbour and Onision. (The former of which got her YouTube account "removed" because of a shit video called "Dear Fat People" she made and now she's crying about getting "#Censored" as a comedian. Like, no, you didn't get censored for your "comedy", you got censored for your bullshit. I mean, if you even got "censored" in the first place.)
Look, I get it, sometimes people bitch and get all up-in-arms over nothing and it's perfectly fine to just dismiss them, but rape jokes, race jokes, sexist jokes, fat jokes, jokes about mental illness and cutting, those are things that can actually hurt people or trigger them. There's so many people who go through ridiculously awful times because of rape or racism or sexism or because they're fat or because of a mental illness and to have a "comedian" sit there and make jokes at the expense of what someone might've gone through and then blame the person who is offended for getting upset about it is just bullshit.
Plus, those things are serious topics and making jokes about them instead of trying to bring awareness and make a change in the world and in people's lives really just makes it worse for the people who suffer because of those things because it causes people to think less of what someone might be going through, or it makes a rapist think what they did is acceptable because people aren't getting mad about it, they're making jokes about it. It's not okay to make a joke at the expense of another person, unless they're all for it and are in on it, and then act like they're in the wrong if they don't laugh and think it's funny.
It teaches the young impressionable fans of the "comedians" who tell these kind of jokes that it's okay to not take these things as seriously or that what they might be dealing with is somehow a joke and it's okay to make fun of it even if it hurts the person suffering.
There are many ways to be funny without being a prick, maybe instead of saying people are "killing comedy" or whatever you consider to be "comedy", you could take a hint and try out some new material.