You do not need a degree to work in the music industry. That's the short answer, however, in reality, it's not really that simple. The answer really comes down to what music industry job you want to get and what your goals are.
Let's look at it like this. If you plan to start your own record label, promotions company, management firm or some other music related business, you don't need a degree to get the job. That's not to say a degree may not help you - after all, having a business education, for instance, will acquaint you with things like writing a business plan, budgeting and accounting. Further, if you are looking for outside investment, having a decent educational background on paper may help convince some purse string holders to take a chance on you. However, many a thriving music business has been started by someone who struggles to calculate a tip after a meal but has an ear for good music.
To that end, many people who run music businesses don't require degrees from the people they hire. What really wows them is experience.
There are some instances in which you do need a degree to get a job in the music industry. Of course you will need to go to law school and pass the bar if you want to work as a lawyer in the industry. Beyond that, some major labels require their employees to have degrees. If that is the world you are trying to get to, then yes, a degree is a benefit.
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 their students in internships during school and jobs after graduation, stay away. Try to find a program that incorporates student run labels and businesses into the learning process. The best way to learn the industry is through experience, so any degree program worth its salt will provide you with plenty to put on your resume.
If you can't major in a music business related field, consider a business degree. If that doesn't float your boat - and I can't blame you - it doesn't matter so much what your degree is in as much as what you do with your time while in school. Get involved with campus radio. Book shows at local venues. Promote releases by musicians who are also enrolled in your school. Do whatever you can to get hands on industry experience, however trivial it may seem, while you're earning your degree. Degrees are great as a fall back, but chances are that outside experience you gather is what is really going to open the most music industry doors for you.
Learn more about music business degree programs: