That is true about the dogs and traffic. There is a ground cover. Purple wintercreeper euonymus. It's hardy to -20F. It is very tough, once established. You can even mow it to keep it growing into the ground, instead of up. Does the area get enough sun?? I did a lawn similar to this situation for a lady, but the real problem was... no sun.
Extreme deep shade , which pacysandra or(japanese spurge) was the other suggestion for shade. Vinca is too fine a leaf for dogs to run on. Another suggestion would be to keep it mulched well. At least would keep the mud and dirt down. or pea gravel with landscaping cloth underneath.