Yes. At your top level forum go to:
Options > Users > Change Permissions
and remove the "Create_topic" and "Reply" from "Anyone" and "Registered" and add it to "Members". It seems that "Anyone" may already "View" your forum.
By default all your sub-forums will inherit these settings, so make sure you haven't made any changes to to the Permissions settings on your sub-forums.
With the settings above, when a non-member attempts to post they will be sent to a screen that invites them to email you with a request to be allowed to post. On the basis of that email you can decided whether to make them a member. Making people "Members" is an entirely manual operation.
Volunteer Helper - but recommending that users move off the platform!
Once the admin for GregHelp now deleted.