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First step in my book try to understand why the co-worker is being rude and defuse the bomb before it goes off by lending polite professional support. After understanding the root of the problem try to offer them professional advise which would be beneficial for the organizational and their professional and career development. Try to help them understand how it may damage the reputation of the company and it's client relations. If they are persistent and do not flick an ear then after observing them for ample time but before a serious incident may occur, you can try approaching the manager/supervisor and update them about it.
If you are the decision maker and despite your constant reminders and advices if this goes on, then you certainly have the option of letting them go. But that should be only in extreme cases and as a last resort.
However a slightly different scenario: The decision of firing the employee can be immediately made depending on the gravity of their conduct and behavior. If the way they treated the clients is over the belt (e.g. abusive) they may be terminated immediately.
Ask once or twice to calm .I’m not his mentor .
CRM have to worry for this concern.