Some people who take offense really do care about the issues underlying. Many people who take offense, however, do so not because they care about the issues underlying, but as a means to control others. They mean to dictate what people may or may not say. They wield discomfort as a club to keep others in line. They are the tyrants of cowardice, and we must oppose them courageously.
1. Prefer a world where people let whatever they mean flow - questions, jokes, serious statements - to a world where people are too uncomfortable to do so.
1A. Apologize to individuals as needed. If a person admits to being personally offended, apologize. Go further. Put out a deeper and clearer statement of what you really feel on the matter. How they react to a sincere offer to discuss soberly the underlying issues will tell you a lot about the nature of that person's objection.
1B. Do not act as if or accuse entire groups of people of being unable to take a joke. If a person claims not to be offended on their own behalf, but rather on behalf of hypothetical others, who "might be offended," inform the person that they are free to be as cowardly on behalf of others as they wish, but that you consider it a disgusting insult to any group of people to assume they so utterly lack that most humanizing of virtues: a sense of humor.
A sense of humor is not the highest human virtue. It may even be the lowest. But it is the one virtue we need most, if we are to get through life with our sanity intact.
If you can't take a bad joke, you can't take a joke. I will suffer having to frown through every bad joke I have to, because I want to live in a world where the good jokes are not stopped by bitten tongues, are not kept shut up behind pursed and worried lips. I want to live in a world where the dangerous questions are not swallowed unasked.
I want to live in a world where serious statements are made. Whenever they need to be.