Standards could help too.
I recently started setting up a phpBB forum for a personal project. Because I wanted to respect people's privacy as much as possible, I removed certain fields like the birthday so that they can't be entered. I disabled private messages to avoid keeping unneeded nominally private data. To contact a specific user, I allowed only emails sent via a form to prevent leaking a user's email address. And I installed an extension to allow users to delete their accounts. I was pleasantly surprised with how easy disabling birthdays and other profile fields were, but somewhat disappointed that allowing users to delete their own accounts wasn't built in yet. Would be nice for forum softwares to have a standard set of features and default behaviors that respect privacy. I doubt many people change the configuration settings I did. (If you have any other ideas for forum admins to make their forum respect privacy better, I'm interested.)
I don't see why a forum should have a birthday field in particular. If COPPA compliance is a concern, just ask if the user is 13 or older at registration.