It really comes down to what music industry job you want to get and what your goals are. If you want to be a performer or a technical type, you probably won't need an educational degree. On the other hand, if you plan to start your own record label, promotions company, management firm or some other music-related venture, it's going to be tough to get banks and other companies to deal with you if you don't have some business training. Or if you join an established outfit, your ability to climb the ranks may be limited.
Whether that means a degree in business or a more general B.A. will vary. But in a business school, you'll learn how to write a business plan, budgeting, and some basic accounting. Admittedly, many a thriving music business has been started by someone who struggles to balance a checkbook but has an ear for good music; for the business to flourish, however, it'll eventually need capital, and without some kind of background or training in this area, why would a bank lend you money or take a chance on you?
Jobs Requiring a Degree
There are some specialized positions that will definitely require a degree. Of course, you will need to go to law school and pass the bar if you want to work as a lawyer in the music industry. Beyond that, some major labels require their top executives and their management-track employees to have degrees. While there are still those in positions of authority at music businesses who don't require degrees—what really wows them is experience—they're getting fewer and farther between.
In any event, it's certainly not going to hurt you to take some kind of basic business training program. If nothing else, such training may help you discover that the business end of things isn't for you.
What Kind of Degree?
If you do go to school, and specifically if you decide to go to school for a music business degree, then make sure the program is up to par. To be worthwhile, a music business degree program has to offer you a lot of experience outside of the classroom. If the school doesn't have a good history of placing students in internships during school and jobs after graduation, stay away. Try to find a program that also incorporates student-run labels and businesses into the curriculum as well, so you're learning by doing all the year.
If you can't major in a music business-related field, consider a general business degree, and find other ways to get music industry experience. Become involved with campus radio. Book shows at local venues. Promote releases by fellow students who are musicians. Review albums and performances or write about music for the school paper. Do whatever you can to get hands-on industry experience, however trivial it may seem, while you're earning your degree.
Given the global nature of the music industry and the intense competition, anything that gives you an advantage will be a good thing. Consider learning the business side of things so that you're well-equipped for any position in the music industry, even one you may not have thought of yet.