The hotels that cater to Westerners typically charge a lot more, and rates are more comparable to hotels in a cheap European city. Theoretically, with them you get a guarantee of food that is prepared with better safety standards. Whether this is practically true I have no idea.
Ironically, the cheapest way to live and eat in India would be to rent an apartment and get a trustworthy cook to come in and cook fresh food for you every day. Your daily bill will be way less than $20. However, setting this up requires a fair amount of legwork and contacts, which (understandably) someone who is looking for a vacation will not find attractive.