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You can organize a meeting with the client together with your subordinate to reassure the client that nothing bad was meant by that action/ word/ e-mail. You also need to go through this case together with the client to make sure it never happens again
Diplomatic in handling this situation, while show emphatic to customer and reassure him/her such issue will not reoccur again.
You have mentioned good staff,your customer's complaint may be wrong,how ever talk to your staff,
even if the good staff is not wrong & customer is your KING,offer apologies implying that your good staff did not mean to insult.
as we know that customers are always right so firstly i will apologize to that customer that will listen what actually happens then i will call my employee to make him or her to understand the value of a customer for the business but i will keep it in my mind that i wont hurt my employees too.
I will ask both of them to submit the report on occurrence and collect separate report from customer in writing. Also try to find witness at the time of occurrence. If failed witness, try to find out both personal background & actual activities in field level. Finally take a decision of punishment for guilty staff.