I use chicken wire on all my runs. ..have never had any problems. If you live in an area where a coon or something will come out during the day and attack your birds, than you may want to consider upping it to welded wire... otherwise, there is no need.
It keeps my chickens in and everything else out just fine. If a dog had half a day to dig under it when no one was home, that may be one thing... maybe keep the birds up if you plan to be gone all day but your birds will absolutely let you know if something suspicious is trying to make its way inside. But, I have never heard of a cat biting its way through chicken wire ...we have lots of strays here and they never even look twice at my chickens... and I have bantams. I do lock mine up at night when most predators are on the prowl. ..and I have welded wire on the windows of my coops.