Sherlock Holmes says time and time again that the status of the clients he serves is less important to him than the intrigue of the problems they bring him. And in fact, we see him interacting with a wide range of social classes, from Peterson the hotel employee, who brings him the surprising goose in "The Blue Carbuncle," to the extremely class-conscious Lord St. Simon in "The Noble Bachelor." So Holmes is pretty even-handed and does believe in equality among the classes. Well, up to a point. Social status is still one of the determining tools for characterization in these stories. It's right up there with gender as a way of working out what a character's weaknesses and strengths will be. So while Holmes may claim not to care about class, he's certainly not blind to it.
Conan Doyle uses dialect, clothing, physical appearance, and emotional sensitivity to suggest differences in social class.
Even though Holmes enjoys exploring the problems of working-class clients as much as high-class ones, he still uses social status as a way of judging people's character.