I don't know exactly what you mean by cheap but the Seaside Guest House is the best known budget/backpacker place in Belize City. Not fancy, but safe, with beds and rooms from around US$15 to $25. Several places are a big step up from that for about US$50 to $70 -- Villa Boscardi, Colton House, Four Fort Street and Biltmore Plaza, among them.
--Lan Sluder
Belize First Magazine
http://www.turq.com/belizefirst/