How does Sherlock Holmes make money?

This is a very interesting question, and one which is not answered specifically or fully in Arthur Conan Doyle's stories. It seems that although Holmes does not actually charge all his clients for his services, he is certainly paid by some of them, particularly the wealthier ones. Essentially, he seems to be able to serve poorer clients because of the remuneration he has received from richer and more corporate or official ones—in "The Final Problem," Holmes says that his recent engagements with the French government and the Scandinavian royal family "have left me in such a position that I could continue to live in the quiet fashion which is most congenial to me"—that is, he has been paid by these institutions on such a scale that he could actually stop working if he wanted to. There are hints throughout the stories that other royal clients in particular have given Holmes money or at least material wealth—he is given an emerald tie pin by Queen Victoria at one point, for example, and Irene Adler and the King of Bohemia present him with rewards for his services in "A Scandal in Bohemia."
What is important to note, however, is that one is not supposed to ask this question—Holmes is a gentleman by breeding, so it is entirely possible that he has some sort of family money; while Watson comes from a professional family and obviously earns his keep as a doctor and from his army pension, Holmes is from a rather mysterious background, and even Watson doesn't know the full ins and outs of it. It was not unusual for a gentleman in the Victorian era to live on "independent means"—an inheritance or similar—and not work, in which case Holmes's income from his work may be only supplementary.

One of the things we like best about the Sherlock Holmes stories is being able to identify with the great detective's lifestyle. He does not have to go to work in an office, and he has acquired such a reputation that he never has to worry about money. He can spend most of his time doing whatever pleases him in his comfortable lodgings, where Mrs. Hudson serves his meals in his own living-room and takes care of all the housework. Holmes as a general rule does not even need to call on potential clients. They come to him, even kings, high-level government officials, police detectives, aristocratic ladies and gentlemen--and he may refuse to accept any client if the case fails to interest him. In the case of the "Adventure of the Speckled Band," Holmes involves himself because of its intriguing aspect and his sympathy for Helen Stoner. He doesn't get paid a cent, but that is of no importance to him. If we readers should wonder about how he manages to maintain such an enviable lifestyle of virtual retirement and leisure, we are shown in other Sherlock Holmes stories how he makes a great deal of money; and so we are satisfied that he can live in comfort and security for the rest of his life. In "The Adventure of the Priory School," for example, Holmes receives a check for six thousand pounds from the Duke of Holderness for a few days of work. That would have been equivalent to $30,000 at the time, and over half-a-million American dollars today.
Sir Arthur Conan Doyle acknowledged his indebtedness to Edgar Allan Poe's detective stories as the inspiration for Sherlock Holmes. Poe's private detective C. Auguste Dupin is also a gentleman of leisure who occasionally gets involved in a case that interests him. In "The Murders in the Rue Morgue" Dupin solves the case pro bono. He explains to his friend, the narrator of the story:
“As for these murders, let us enter into some examinations for ourselves, before we make up an opinion respecting them. An inquiry will afford us amusement,” (I thought this an odd term, so applied, but said nothing) “and, besides, Le Bon once rendered me a service for which I am not ungrateful. We will go and see the premises with our own eyes. I know G—, the Prefect of Police, and shall have no difficulty in obtaining the necessary permission.”
Dupin is not concerned about money. He is looking for "amusement" and wishes to help a young man wrongfully accused of the gruesome murders. But in "The Purloined Letter" Dupin is told by Monsieur G., the Prefect of the Parisian Police, that a huge reward has been offered for the recovery of the letter:
“Why, a very great deal—a very liberal reward—I don't like to say how much, precisely; but one thing I will say, that I wouldn't mind giving my individual check for fifty thousand francs to any one who could obtain me that letter.
At the story's conclusion, Dupin collects that share of Monsieur G's reward, which should be sufficient to keep Dupin in comfortable indolence for many years.
In both "The Adventure of the Priory School" and "The Purloined Letter," the intention of the authors seems to be to assure the reader that their detective-heroes are able to maintain an enviable lifestyle without having to hold any kind of regular job. Both have complete freedom from work and freedom from financial worry. We would all like to be like them, and that is one of the reasons we find their stories so interesting that we can read them over and over again, even when we know how they are going to end.