Youth Hostel Association Old Steine, 5 min walk from Dome. 180 beds, across 51 rooms.
Kipps Backpackers Hostel Grand Parade, 2 min walk from Dome.
Hostelworld also lists other options in the city, with an availability search tool and reviews.
Other hostels outside Brighton:
Lewes YHA 66 beds, some camping. about 30 mins from Brighton by train or car (5 min walk to Southease station).
Truleigh Hill YHA 52 beds plus camping. 40 min drive from Brighton.
Eastbourne YHA 33 beds, about 50 mins by train or car.
Littlehampton YHA 35 beds, about 1 hour by train or car.