Basic Auth Exceptions: favicon.ico

<Directory /var/www/mysite>
          SSLRequireSSL
          AuthName "Authentication"
          AuthType Basic
          #AuthBasicProvider internal #LDAP
          AuthBasicAuthoritative off
          #AuthzLDAPAuthoritative off #LDAP
          AuthUserFile /dev/null
          Require valid-user
          SetEnvIf Request_URI "^/static/icons/touch-icon(-ipad)?(-iphone)?(-retina)?.png$" allow
          SetEnvIf Request_URI "^/static/icons/favicon3.png$" allow
          SetEnvIf Request_URI "^/static/images/apple-touch-startup-image(-([0-9]+x[0-9]+))?\.png$" allow
          Allow from env=allow
          Order allow,deny
          Satisfy any
  </Directory>

Replace those regexes with where the <link>ed icons are.