Smart-mouthed colleagues normally struggle with their own insecurity and believe that sarcasm helps them look better to others. I would normally put myself in the smart-mouth's shoes and identify myself with that person. When they sense that l understand, it helps bring their guard down, preparing the way for an easier confrontation.
I would confront smart-mouth individually rather than address the team as a whole. Smart-alecks may not be aware we're referring to them when addressed as a group and the non-jokesters may not appreciate the association with the guilty party. Smile, be calm and give an example of a specific sarcastic remark and how that made one feel when one talk to the offender. I would continue to do my job with confidence and show kindness to the smart-mouth. Perhaps the smart-mouth will learn a thing or two from my own professionalism :-)
The clear problem of the outlawing of insult is that too many things can be interpreted as such. Criticism, ridicule, sarcasm, merely stating an alternative point of view to the orthodoxy, can be interpreted as insult.~ Rowan Atkinson (Mr Bean)