In short, it depends.
I've seen many stories pop up on my Yahoo news feed that spout all the reasons a degree in any of the arts is pointless. And while some of their reasonings are valid, I don't think they are wholly correct in the pointlessness of an arts education.
Can you succeed as an artist without an arts education? Yes.
Should you give four years of your life and spend thousands of dollars on a piece of paper that says "hey, you are certified to do art now?" Maybe.
Will people take you more seriously if you have a degree? Not necessarily.
That isn't to say that a degree in the art form for which you are most passionate can't be beneficial.
1. Learning from professors can make you more confident in your own knowledge.
Professors have spent their careers perfecting their craft in order to share with others what they have learned. They have gone through their own trial and error processes and pass that knowledge on to you, either so that you don't have to go through your own trial and errors or so that you can do the same process but work through it for a different outcome. In either situation you come out more knowledgeable and thus more confident in your own work.
2. Professors can help you find "your" art.
But professors aren't just there to teach us stuff about what they already know. The best ones are there to help us learn about ourselves. I went to college for a theatre arts degree. I took a playwriting class, for which I had to audition, that was set up as a sort of round-table where we all workshopped our plays in order to better them. I took everyone's suggestions to heart and eventually it turned into a play that I neither liked nor recognized as my own. The professor encouraged me to go back to a draft where I was still in love with my work and encouraged me to listen to my own artistic desires over the well-intentioned, but possibly wrong, suggestions of others. This suggestion of listening to my own artistic desires has been something I've carried with me through the rest of my works.
3. An arts degree enables networking.
A lot of the people who work at a university aren't there just because they couldn't make it out in the real world. A lot of them are there because they have been successful in the real world and were sought after in order to teach about their successes. That means they know a lot of people on the outside, people who can help further your own arts career. At a university you have all sorts of experts at your fingertips, whose salaries you are helping to pay for; take advantage of that, find a professor you really connect with and use that relationship and their know-how to get in contact with other like minded artists in your post-education art career. It takes out a lot of the leg-work later on.
4. An arts education can fuel your passion even more.
Or in my case, make it even harder to pinpoint one specific thing you want to do. But in any case, surrounding yourself on a daily basis with like-minded people can only be a benefit than a hinderance. It's why there are conventions for all sorts of things from techies to porcelain cookie jar collectors. When you are surrounded by the thing you love you want to work on it more. You learn new techniques that you can't wait to incorporate into your next project. You meet someone in a class who has this brilliant idea that you want more than anything to be a part of and before you know it you two have produced the next big indie film. And in the end if the degree you went in for isn't the one you came out with you still learned what your passion really was.
Going for an arts degree is a decision that should be considered very seriously. Many will argue that an arts degree will not guarantee success or even a viable job. True. But that can also be said of any degree. If everyone goes for a practical degree pretty soon we will have countless numbers of freshly graduated accountants but not enough jobs to accommodate all of them. Nothing is guaranteed, but hard work and passion can fuel any successful outcome, no matter whether you choose the art degree or the degree-less path.