Blanca and Carlos, a wonderful Ecuadorian couple, will rent you a private room in their home (there are six bedrooms, I believe, and two baths). Breakfast-only is $10 a night, $11 breakfast and dinner, or $12 full room and board. It's available for long- or short-term stays, though longer stays are the norm. The price include laundry service and airport pickup.
The family speaks mostly Spanish, but have been hosting foreigners for years and are VERY patient with beginners. You will probably be invited along on family activities as well.
It's about a 10-minute walk from the tourist-prone Mariscal district, meaning it's a safe and quiet neighborhood (La Floresta), but still convenient to the restaurants, internet cafes, Spanish schools, and nightlife of Mariscal (aka "Gringolandia"!). The old city is 20-30 minutes away, by taking a bus (or walking) to the trolley ($.25 each).
This is an ideal homestay for a student, volunteer, etc. They treat you like family, but don't mind if you stumble in at 3am either. The rooms are simple and clean, and Blanca's cooking is tasty and pretty healthy as well - a great intro to Ecuadorian cuisine, especially at lunchtime.
There is a dog, but she's pretty friendly and is hardly ever inside the house.
Some of the rooms are lacking in privacy, though, so you may want to see your room first. The house is a bit chilly, but that's pretty much the norm in Quito, where heating is unheard of.