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.