Ask your web hosting provider about access logs - they'll contain all the data you'll need... and lots, lots more. Probably too much, in fact!
If you're worried about a security problem... well, first things first - make sure HTML and the IMG UBBCode is off. Switch the board into members-only mode so that nonregisterd members can't view the board. And if you haven't done so already, turn on member moderation. You can check on the IP that the user registered with in the registration queue.
When it's released, I would also advise upgrading to 6.2.1 - it includes an admin-only view to Recent Visitors that includes IP addresses.