Heres my question...

I would think they can.  I walk my dogs at the college where I work, but they must be on-lead and they are not allowed in any buildings or at the nature preserve and it is enforced by campus security.   We run them off lead on the baseball fields, but we got special permission from the coach, who got special permission from the school (she runs her own dogs there).
 

I think they can defiantly enforce this if they choose because there is a sign clearly posted, and it's school property not public. I wouldn't take her there, at least not during school months.

The beach that I live on doesn't have a fence either, but there are signs posted at every entrance "No Dogs Allowed June-September", so even though I would clean up after Apollo and keep him leashed I don't bring him there during the restricted months.

They can enforce it. It isn't public property. But they can also chose not to enforce it if you are a responsible owner they might not care. 

 I think its a classic case of somebody ruined it for everyone else.  Obviously somebody has had a problem with dogs there at some point, likely the groundskeeper.  Probably somebody doesn't clean up after their dog, or lets their dog run loose.  It sucks, but I see signs like that going up all over the place. 

huskymom

 I think its a classic case of somebody ruined it for everyone else.  

So true.  All it takes is one person who goes there regularly and doesn't clean up to get the "no dogs allowed" signs put up.

Joyce