Hide container when there are no visible children

I've come across a case where I have a Members menu container with children menu items available only to logged-in users. It would be great if the menu container offered the option of simply not appearing when anonymous users visit and are no child items they are authorized to see. If I used a regular menu item, I believe that behavior is automatic, but this site structure really doesn't lend itself to a clickable Members menu heading.

Thanks!