I'm not entirely sure if you're saying "toughen up" is never the answer, or "firing" is never the answer.
Assuming the former: It is entirely possible to misinterpret friendly ribbing or banter as something far more malicious. To assume the worst of someone who genuinely means no harm, and to suggest they should be punished, censored, and shunned. "Toughen up" is perhaps not quite the right phrasing here: It's asking people to perhaps extend the basic human dignity and respect you request of others, to others. Where, perhaps, simply asking them firmly to knock it off, explaining you are finding their jokes hurtful, and giving them the chance to change their behavior, is perhaps more appropriate than forming an internet lynch mob based on assumptions of malice. Treating the other person as a person capable of mistakes or being misunderstood, instead of a gremlin to simply assume the worst of.
Assuming the latter: Sometimes you simply can't work through the problem. They may be a toxic individual - they may simply be driven to toxic behavior at this job (perhaps they hate it) - perhaps it's simply a matter of deciding who you're going to keep between two people who simply cannot get along with each other, to the point of causing problems for the rest of the team. Assuming you can fix their issues when they can't isn't treating them with dignity or respect. Leaving the situation to fester and drag on isn't treating the rest of your team with dignity and respect. Firing one or both of them might not be treating them with dignity and respect either - but you've only so many options. But just because you're firing someone doesn't mean you can't at least strive to do so. Wish them well on their next venture, and mean it.