Figure out who your allies are. Will your boss back your efforts to improve the tone? Find other people who resist the sarcasm trend and consider whether you can work together to bring about a change.
Know where and how you can influence the tone. Also, know your threshold for tolerating the negativity. You deserve to work in a place that satisfies you on a variety of levels, and there’s no reason to submit yourself to an environment that causes too high a level of discomfort.
The place where you’ll have the most influence is with your team. If your expectation is that sarcasm is inappropriate, then state that expectation. Tell the team as a whole, and talk with each person individually. Expect pushback; this will be a hard habit for them to break.
You’ll need rewards and consequences. What will people on your team get out of it if they change their tone? Even small things can motivate, and the change to a positive tone can be reinforcing on its own. And what are the consequences? Do you have the authority to let people go if they can’t get with the program? If so, if people are resisting, let them know that this is a consequence; it would only be fair.