There are affordable family-run hotels in Paris, but do not expect the same comfort as in a three star one: full breakfasts, private bathrooms, elevators and phone are rare if you wish to pay less than 50 € a night.
Also avoid hotels located in the least secure neighbourhoods or Paris (Barbes, Belleville, Pigalle, rue Saint Denis).
Here are some examples of family-run hotels in the central arrondissements of Paris.
Hotel Henri 4
25 place Dauphine