Stage file proxy - website under shield
See module's INSTALL.txt file:


  If the site is using HTTP Basic Authentication (the browser popup for
  username and password) you can embed those in the url. Be sure to URL
  encode any special characters:
  
  For example, setting a user name of "myusername" and password as,
  "letme&amp;in" the configuration would be the following:
  
  $conf['stage_file_proxy_origin'] =
  'http://myusername:letme%26in@example.com';


Another way, if you have a static IP, is to bypass the Apache auth for certain IP addresses.