I know The University has apartments for married students but am not sure about the process there. There is now an amazing variety of student living communities in the private sector too, most offering great amenities and free transportation. Again, I don't know how much any of these are specifically geared to graduate students but since WVU has decided to focus their efforts on increasing research and graduate education I would anticipate that there will be growing housing opportunities there too.
You can check the Sunday Dominion Post newspaper (it's online too) for many offerings, but you'll want to hurry because Morgantown is a high demand area and quality housing fills up quickly with folks reserving places for the following school year. I'll include links to some of the student housing communities.
The Ridge | West Virginia University Student Housing, Morgantown College Housing, West Virginia University Off Campus Housing
Pinnacle Height Apartments - Morgantown WV 26505 - Apartment Unit For Rent
Best Morgantown West Virginia Apartments | WVU Off Campus Housing
Copper Beech Townhomes apartments in Morgantown, West Virginia
morgantown apts/housing for rent classifieds - craigslist
Apartment for Rent - $700 - Town house style...near Evansdale campus (Morgantown, West-Virginia) - PadMapper
This place isn't near Evansdale, but you can get there free on the PRT or via local busses...
Mountaineer Place Apartments Rentals in Morgantown, WV
I'm just enclosing the above as examples and there are lots more. Do a Google search and you'll see, or better yet visit for a couple days and look around. I'm sure you'll find something you like.