To make private forum first remove the "View" permission from those in both the "Anyone" and "Registered" groups, then even if a stranger registers they will not be able to access your forum.
Make sure those in the "Members" group have the "View" permission enabled and add all your students email addresses to the "Members" group, then when they register they will have access to the forum, while others don't.
Volunteer Helper - but recommending that users move off the platform!
Once the admin for GregHelp now deleted.