The board I am a member of is pretty quiet and most of the threads come from its creator. I think she pulled from a pretty small pool though and it is a board based on a pretty specific category of WebMD members.
We have not had one single incident of Anon postings or things getting out of hand on that board since it was created. The thing you have to worry about though if you have some hard-core trolls stalking you that are determined to cause trouble is that they can always get a new username to gain access to a private board too. Understandably it will probably happen a lot less often than trolling on a public board but it is still possible.
That would also be a question for WebMD administration about being permitted to remove invited members who are causing problems (although I don't know how you would go about this with Anon postings).
I think I would recommend a private board, especially considering the problems that have been touched on here from other boards and that I have seen glimpses of around here but whoever chooses who will be invited will have to beware of trolls, especially with members who just recently joined.