We find Invisionfree is fine - they have boards which can have manage 1,000's of members.
However, its not perfect (hell no board system is). The new software has just been released and is called Zetaboards.
This is pretty good from what we've seen so far, and it does have some nice new features our board members love!
Its not long out of beta, so again there are one or two issues cropping up, but overall I believe its the best Hosted platfrom now out there.
Proboards are good - But I've found one or two annoying problems with them that just didn't allow us to have a good smooth experience.
I also understand people are using Runboards - And seem happy enough with the set up there.
So I would suggest go to each location, register a test board - Have a quick look and test with all of them, ask here if you need help (we've got people here who have had experience now on all of these systems) and we'll see if we can't give you some further guidelines.
So, here are some links: (I've omitted Invisionfree for now and just included Zeta).
Zetaboards - The new upgrade from IF.Proboards - Another hosted Forum company.Runboard.com - Another option *used by Queeny*