Forums that require registration before the content becomes viewable does affect SEO because popularity plays a role in the ranking formula. If your forum requires people to register to view your content, it's likely that they simply won't take the time and effort to do this, and they'll look elsewhere. This means less traffic on your forum.
Open forums, on the contrary, allow all internet users to have access to content. This, in turn, increases the searchability of the content because open forums generally have a higher engagement rate, and a high engagement rate positively impacts SERP rank.