Most theatres do not make money off their ticket sales and are dependent on the kindness of strangers. Even commercial productions are fraught with failure, as only 30 percent of Broadway shows make back their money. Producers pass on this problematic financial model to their employees.
Early in their careers, directors often work for free or a small stipend. However, once they start working consistently on a professional level, most will join SDC, the Society of Stage Directors and Choreographers (yes, directors are put in the same category as fancy prancers). This is the Director's Union and will ensure a director is paid appropriately and receives health benefits if working regularly.
At the union level, directors should not be making less than $500 per week, although exceptions can be made if the directors want to invest their time developing a show that will make them money later (good luck, sucker).
For a typical four-week rehearsal process at a recognized regional theatre, a fee of $5,000-$20,000 can be expected, depending on the size of the house. Remember that this has to cover preproduction, all those hours spent pulling out eyebrow hairs one at a time while considering if Ophelia's dress should be peach or salmon. This fee must also cover all the time spent sitting on a couch that smells like tears, waiting for the phone to ring.
By the time you're directing on Broadway, you should expect to be making ten times this amount, or $50,000 and up per production.
However, where the real money is to be made is in royalties. Directors who work on initial productions may negotiate for a percentage of the profit. That means if you directed the original West Side Story, you continue to get paid every time the movie is shown on TNT, every time a Sondheim fangirl buys the soundtrack, and every time someone jumps in the air and snaps their fingers.
The other route is to become an Artistic Director of a theatre and hire yourself. Then you can direct until you die. Literally die, because nobody leaves that position until that happens. This job involves a lot of administration and fundraising (read: one hundred and one ways to stick your nose up a behind) that may or may not appeal to you. (Hey, no judgment here.)
Many directors pick up extra cash teaching college or directing industrials. An example of an industrial is when an oil company wants to throw a party showing off their new machine that will magically clean up that oil spill, complete with song and dance. For this, directors can earn a large fee in exchange for part of their soul.