Great question. First thing, of course, is that at a public institution where the First Amendment applies, even "offensive" speech is legally protected speech. So even a joke that goes badly wrong and hacks off half the campus is still not a lawful basis for the school to de-fund the paper, fire the editors or otherwise retaliate.
A couple things. First, calmly (but quickly) assess how badly you screwed up. If you did something really seriously wrong, then you'll want to get out in front of it and explain, apologize and (if warranted) take some action against the offender(s) internally. That's key -- you want to show the public and the college that the newsroom has its own accountability system so that the college doesn't feel that it needs to step in and impose external controls.
Remember of course that there is no legal right not to be offended, so there's no legitimate legal exposure for a story that is merely offensive (as opposed to libelous, i.e., making false factual accusations against identifiable individuals). So while you may suffer some damage with readers and advertisers, you shouldn't be at any real litigation risk.