It depends what you mean by cheap. $ 500/month and you'll do alright.
There's Caribbean Dreams which is reasonably close to Cross Creek, further out ( on the point ) there's Sandstone.
I stayed at a Maggie's Place which is across from the BICD, $450/month for an awesome one bedroom apartment. Solid concrete so I was quiet, no bugs, not even a gecko in the place, there's no way for them to get in, hot water shower, aircon, designer bed linens, spotlessly clean, patio with BBQ, washer and dryer ( free ), TV, a proper internet connection, and a fully equipped kitchen, The place is only five years old, but I would have thought it was brand new
The down side, the cost of electricity. $0.35/KWH meaning it costs about $1/day just to keep the lights/TV/fridge/hot water on. Fire up the aircon and easily double that.
There's and American couple on the island, Mike and Jeanie, who run an apartment finders service. Link up with them and they'll sort you out. See their ad outside the Bundu Cafe or at Deep Blue Divers.