Qualifications
The primary qualification to be a producer is simply the ability to convince people to give you their money (without pointing a weapon at them or threatening their families). Producers are social animals first and foremost; if they didn't get into producing, they might have gone into sales or politics.
If you have an Arab oil magnate as a favored uncle, that'll do ya better than any film or business school degree. However, if you don't….
A degree in business or film studies can be useful, especially if you want to intern for a production company or get a job as someone's assistant (source). Producers need to have a keen understanding of how a movie works, as well as all of the avenues that a movie has to make money.
They need to know how the money flows in, from theatrical distribution through DVD sales to television and the Internet. They need to know how to structure investments and create solid budgets and business plans. Also important is a knowledge of marketing and an understanding of the legal aspects of filmmaking. They also need to network like crazy. And then network some more.
Remember: A producer's job is to essentially create money out of thin air. Without the use of magic.
In a nutshell, although you don't need a PhD, it's not a job everyone can do.