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Hard to say since we don't know where you live or what kind of predators you have.
But, if you lock the chickens into a well ventilated building at dusk, likely not.
If the coop is open into the run, then yes. You would then need some type of barrier to predators getting into the run.
If so, the hardware cloth or woven wire barrier doesn't need to be buried straight down. It is just as effective as a skirt extending out away from the run an inch or so underground.
Some places on the planet have few predators. I have raccoons, opossums, mink, weasels, fox, coyote, dogs, cats, hawks and bobcats. Bears/cougars are moving this way.
Edited by ChickenCanoe - 5/6/16 at 1:29pm