NY to Hong Kong and Air Asia to Bangkok about $1000 total
Don't know how much this will save you - fares start at about $870 from NYC to HKG in the summer using a Kayak flexible search. From Hong Kong there are direct flights on Air Asia to Bangkok starting at $60 USD or so each way on the dates I looked
Hong Kong itself is well worth a visit, and I've found the hostels in Chungking Mansions to be clean and safe option despite its reputation. (A private room starts at about $20 USD/night.)