There are two steps to do:
1st - Select in the customer's file the options to block all communications.
Inside the customer's file. Go to Additional Information section and click on all the options related with the communications the customer doesn't want (email, phone, sms...).
2nd - Check and remove the customer's email account from the newsletter list.
Go to Website and Networks → Users subscribed to newsletters and search the customer's email account on the search bar. When you find it, press 
