Qualifications
Theatre school is a great way to look like you know what you're doing. Common degrees are a BFA in drama or its less-popular sister the BA in theatre arts. For the scholarly artist, an MFA is an advanced choice, and also an easier way to get one of those incredibly sweet university program director/part-time teacher jobs (source).
In those places, even students who don't come to see the plays (usually most of them) are still technically paying for your production.
While school definitely helps, there's always the exception. The most important thing is to learn by doing. Producers will only hire you if they know your work. Fortunately, there are some valuable opportunities for early-career directors. The Drama League and Lincoln Center Theatre offer such opportunities for you to direct weird freaky stuff before you've proven to the industry you're not insane (well, mostly).
Once you're ready, joining the Society of Stage Directors and Choreographers will give you some credibility. This also ensures that the theatre in Podunk, New Jersey, doesn't stiff you.