Send weekly news, update content, post interesting posts and question which make other to participate there in the forum.
I regards to the adding content. The more frequently your site changes, the more frequent the visits will be from Googlebot. There are some sites that get crawled multiple times a day because of the changing content. So, if you are constantly adding new fresh content (which you get from an active forum), then that will give you an edge when it comes to your SEO work. So really, the short answer is yes, the more people posting in the forum, the happier the spiders are. Also, if you get enough people signing up, then you may get some bonus links from them telling people about the forum.