There’s a whole case on how customers will interact more flirtatiously with customer support representatives with a female avatar. Given that, it should come as no surprise that there is no shortage of customers willing and able to cross boundaries when it comes to support team members.
If a customer asks for a date or suggests something inappropriate, you should gently tell them so, and then also notify a manager so that they can keep an eye on the situation. You do not have to be direct or aggressive with the customer, but do let them know that there are penalties for behaving inappropriately. Here’s an example:
Hi there,
Thanks for your response. While I am going to respond to your request, first I wanted to address your comment about [whatever they said that was inappropriate]. This is a professional conversation, and I think it’s important to continue to focus our efforts on getting your issue resolved. In the event that we aren’t able to work together to resolve the issue without further inappropriate comments, I’ll have to have you talk with my manager instead.
With that being said, [insert resolution to their issue here].
I hope that all helps, but please let me know if you have any other questions or concerns.
This keeps the conversation professional without being too accusatory, and also lets the customer know that there will be escalation if the issue persists.