First, you want a grass that is tough and suited for that area.
Assuming it gets sun, my choice would be a good turf type tall fescue.
Normally, fescue is a clump type grass, meaning the plant grows in one
clump and won't send out rhizomes to spread. However, there are now
some varieties available that have been developed to spread. They
still don't spread as readily as bluegrass, but they will spread to
fill in bare spots over time, which a regular fescue won't. RTF is
one variety, you can search for it on the web. I've also seen it at
HD. That could be a good choice. Of course as others have pointed
out, if the dogs are on it a lot, there maybe areas that you really
If you go that route, I would probably wait till fall, which is optimum
time, then kill what's there, and use a core aerator, then a slice
seeder to establish the new lawn.