Ideas of what to do with muddy, torn up grass?

https://www.pinterest.com/explore/dog-friendly-backyard/?lp=true

I would put some pavers down in the path that they run. You could leave space between the pavers and the fence for some greenery, but there is a chance they will get trampled.

dig a canal wide and deep enough to lay drainage pipe with predrilled holes. Purchase pipe at Home Depot or Lowe’s, a building supply company. Lay pea gravel or garden stone both under and over enough 2 or 3 inches deep to meet top of ground. Make sure it is wide enough for dogs tracks.

Install a couple rows of concrete stepping stones.

Could you lay some old used carpeting in their runway area?

Charmaine's answer above is fabulous, but I would add that in order to keep the gravel surface in place, you'll need to edge it. My choice would be a steel edging that you pound in. It will not allow the dogs to track the gravel across the yard, thus saving you from lawnmower blade replacement.

You’re lucky if the dogs are not digging under the fence. But I know what that muddy ground is like. I would let the dogs have the foot-wide strip of earth next to the fence. Stake down a double thickness of artificial grass. Or you could use leftover shingles, gravel that won’t hurt your dogs’ paws, or a thick layer of sand (which would likely become a poop area, but can be sprinkled with more sand. Or you could go hard surface with a concrete walk. Long term, you could plant prickly shrubs, but you would deprive your dogs of a lot of playtime. ☺️

You could put a stone pathway in using stepping stones - there are some cute ideas on here for making your own stones using large leaves, or putting glass beads in the cement before it sets - you could even put your kid's hand-prints in the cement (& have them paint them after it sets - what kid doesn't love helping with a project?) or your dog's paw prints!

I was also thinking of making the area at an angle by using fencing so they can't run back and forth. Not sure what to use.

Put some large bark mulch down. Or regular mulch with stepping stones / individual pavers.