Just like the most of Thailand, Chiang Mai can be as cheap as you want it to be, and as expensive as you can afford it to be. There are loads of ways you can enjoy Chiang Mai on a small budget.
This is probably the most important way of saving money in any city of Thailand, bargain as hard as you can before you agree upon a price. Chiang Mai is a great location for tourists to shop, which is also the reason why shopkeepers inflate the prices. Be courageous and object to the price that the shopkeeper starts with.
Eat street food:
The street food of Chiang Mai (again, like most of Thailand) is not only hands-down delicious, bus is also the cheapest option of food you can get. You can get filling serving of noodle soup, fried rice, or pad thai, and barely pay more than 30 baht for it.
Use songthaew for transportation:
Paying 20 baht for anywhere in the city is the cheapest transport option that you can find. They run all night, and are a decent way of getting around. And the most important perk, they save tons of money.