If you want to stop sending messages to a particular customer, you can add their email address or Order ID to the Email Blacklist. This will prevent any further emails from being sent to them through the Communications Center. Customers will be also be added to the email blacklist if you have activated the "Auto Blacklist on Negative Feedback" in your Message Settings.
You can access the blacklist by clicking Settings under Communications in your left navigation and then clicking on the Blacklist tab. Click the green "Add Customer to Blacklist" button to add a new customer to the blacklist. Input their email address or order-id into the text field, then click "Add." You should now see that their email address has been added to the blacklist. Once the correct address appears here, they will no longer receive messages from the Communications Center.