I prefer a covered run just for the added hawk protection. I have a neighbor who does not cover his runs, but he doesn't seem to mind the occasional bird eaten by the hawks.
My birds free range every day so I only use the run if I'm not home or gone on vacation.
I made an apron around my run using chicken wire that is buried in some gravel we had laying around.
I've only noticed something tried to dig into the run once but that was because a lot of food had spilled in one corner and I didn't clean it up so it was left over night.
Whatever critter was out there tried to get to that food but could not get thru the wire and gravel.
I would also build a run that you can stand up in; even if it's only 5' tall. Makes it easier to maintain.