I got this response from Support:
To allow challenges from only friends, you can set this up by hovering your cursor over 'Home' and select 'Account' then 'Manage Online Settings' which should take you to this page: http://www.chess.com/home/echess_settings
Then, click on 'Friends OK, friends can challenge me no matter what'
Then near the bottom of the page click on 'Stop! I'm not available for challenges or Tournaments.'
Once you are done click 'Save Settings' and you should be all set.
How do I play with Friends Only? I've have just joined and now have three unsolicited games all of which I will apparently lose if I don't respond. At this point I would like to play games with friends only. How can I do that?