Cusco is chock full of hotels and hostels for every budget. There are over 2,200 options for lodging in Cusco, not counting nearby areas.
Cusco has both luxury and budget options in the center near the Plaza de Armas, though prices do tend to drop as you move away from the center.
If you want a quieter, pedestrian area, check out San Blas and Lucrepata. If you’re into nightlife, look at Procuradores and Tecsecocha.
Staying right in the centre isn’t essential in Cusco, which is relatively compact, easily walkable, and well-served by buses and taxis. So there's no need to really agonize over which neighborhood to choose.