I spent five years working for a campus-area landlord, and I can certainly attest to what others have said, that the closer you are to the university, the more hard-partying students you'll find as neighbors.
You definitely want to avoid the State Street area, Langdon Street (a/k/a Frat Row), most of West Gilman, West Gorham and West Johnson Streets, from about the zero block to about the 2000 block. That's the area that tchemgrrl describes as being between the campus and the capitol, and she is entirely correct in describing it as full of undergrads. The area around the stadium is also an area that I would avoid; you'll find people having tail-gater parties on your front lawn starting at 7:30 a.m. on game days.
If you're looking at a map of Madison, look for the very center of town where the capitol is and draw a line from it to each of the lakes. You've just identified where Wisconsin Avenue goes up to Lake Mendota and where Martin Luther King Avenue goes down to Lake Monona. Anything east of there is going to be progressively less and less undergrad focused; there'll still be younger students there, but the hardest partiers want to be as close as possible to State Street and Frat Row.
Most of the rental housing stock in that area has seen better days, to be brutally honest. The newest buildings you'll find are at least 4 or 5 decades old, and there are plenty of century-old homes that have been split up into a bunch of small student apartments. Some are well kept, some aren't; a lot depends on the landlord, of course.
You should definitely check in with the folks at
Tenant Resource Center in the early stages of your hunt. The Tenant Resource Center operates as both an intermediary service between tenants and landlords, and as a sort of Better Business Bureau for landlords. They can tell you whether a property owner takes good care of his buildings and treats his tenants right, or if he's a slumlord. The city of Madison is extremely tenant-friendly; there are all sorts of ordinances on the books that protect tenants from predatory landlords, and the Tenant Resource Center can tell you all about them.
Also, another site that you'll want to visit before you sign a lease is this one:
Average Energy Use and Cost for Residential Addresses . You can learn the highest, the lowest and the average energy cost for any address in the city there, and if your lease requires you to pay for your own heat or electricity, you'd better do your homework before you sign on the dotted line. Some buildings can cost you more for heat than you pay in rent.
Good luck to you in your hunt! My sister came here for grad school about a decade and a half ago, and in fact rented an apartment sight unseen from Baltimore, where she did her undergrad work at Hopkins, so I know that it can be done successfully. Just keep in mind that nothing locks you in to staying at the same apartment for your entire grad school career; if you decide you're not fond of the neighborhood or the building, you can always move in a year. We'll watch for you later this summer!