MyBB Community Forums

I have a staff forum that is only visible to staff.

Will spiders be reading, indexing, and archiving this particular forum for their search engines? If so, is there a plugin or modification I can make to prevent this?

Sorry, I'm really not sure where this goes as I'm not sure what's available and what's not.

You can disallow page in robots.txt
http://www.robotstxt.org/robotstxt.html

You can exclude the groups "unregistered" or "bot" (if you have that) from forum access, that should be enough.