How do you handle customer complaints over your social media Facebook or LinkedIn page?

By getting back to the customer immediately and be positive regardless of whether its true or false apologize on behalf of the company and try and solve whatever the problem was . ensure your complaint is satisfied with the answers and he or she feels the problems have neen adressed completely with zero% doubt

Prompt response to customer of social networking really encourages the customer that wow!! what a quick response. I ve used this skill while managing social networking page of an institution. Students really found it extra beneficial and highly appreciated the work. After all its its better to leave a message of FB than going to their customer service for query 

I agree with what Miriam has said.

Answer as soon as possible and try understanding the customers concern. Many people tend to delete or remove such comments from their page which is not at all a good practice. The best way to deal with such issues is head on and ensure that the customer is satisfied/happy due to your approach towards addressing his/her concern. 

By replying in positive or neutral tone and by sending message in customer mailbox assuring to rectify the issue on asap basis!