You need to go into the "User Accounts" tab on the administration side and edit the "roles and policies" for anonymous users.
As a default they get to see (read) all content in "Section (standard)". You need to edit that policy so that they can only read what you want them to. ie. choose everything other than your new class.
Oh, other ways ... you could put the form in the "restricted" section, or maybe wrap the form part of the template in an {if} routine that checks whether the user is logged in. See the one used for checking user logins for comments.
Though, with wrapping the form in an {if} statement, I wonder whether there would be a slight risk of someone who knows the form field names posting the info anyway ... someone else will have to answer that one.
You mean you only have access with admin users? well, this is how it works by default, but you can configure roles to let other type of users work with that section.