That's an interesting question. I'd say the first indication is in the list of characters, which lists her father as "a Senator." Soon after Iago and Roderigo wake him, Brabantio answers that "My house is not a grange," implying that it deserves treatment above the level of the peasant. When he is upset, Brabantio calls out, "Call up all of my people," indicating he has many servants (and is therefore at least rich, and probably noble).
Finally, when he hears that the duke is awake, he refers to his fellow brothers of the state, indicating he's from the ruling class.