In short, no. A degree isn't absolutely necessary for success. Just like many other career fields, your expertise and experience may be able to help you stand on your own two feet.
However, many writers do have a formal educational background in Writing, English, Advertising, Communications, Journalism, or higher level degrees such as an MFA or a Master's in Professional Writing. And you may have to compete with those writers.
So, unless your writing is strikingly flawless, it will never hurt you to pursue your craft academically. There's a wide range of options available to fit most needs.
Where to Start
Interested in brushing up on the basics? A good place to begin exploring that option is your local community college or even the continuing education program at a nearby university. These classes most often cover the basics, or they'll focus exclusively on very specific niches.
Online Writing Education
Outlets such as Gotham Writers Workshop and WritersOnlineWorkshops offer recurring online classes on many writing subjects. All you have to do is log on. Unsure about the quality of these offerings? You can also check for online courses from your local college, or take a look at the online offerings from institutions such as the University of Wisconsin-Madison or even UC Berkeley.
So You Want a "Real" Degree?
It used to be that writing-prone undergrads pretty much had to choose the English route or the Journalism route. But what about those who want to write simply for their love and passion for writing? What about freelancers? What if you don't have a novel in you?
Little by little, the BA in creative writing seemed to start holding its own and seemed to become a little more prevalent. But that didn't help all of the writers out there. It seemed that non-fiction writers were still pushed to the Journalism standby.
But, (finally) the strict "Writing" degree emerged, and writers were able to pursue a program that covered all the basics, without the need for a hardcore genre or niche designation (yet).
Of course, if your talent and passion runs that way, there are still those great standbys that will indeed take you places: English, Journalism, Communications. Research thoroughly!
What to Do If You Already Have an Undergrad Degree
Time for the next level? Of course, there's always the vaunted Master's in Fine Arts (MFA) with a creative designation. But, if your writing doesn't lean that way, you still have grad level choices.
For example, universities such as Carnegie Mellon and Michigan State University offer a Master's in Professional Writing, which focuses on rhetoric, communication design, and non-fiction uses of writing. Chatham University offers the same degree, only 100% online.
Another option may be a Master's in Publishing, which is currently available at Portland State University, among others. Online degrees are available, too.
No matter what your goals, a formal education could be a great boon to your writing. In addition to the forced writing practice and great contacts you are sure to make in such an arena, you're also likely to hone your craft in a short time. However, for writers concentrating on certain outlets, and who possess talent, the lack of a degree is not likely to impact you as much as you might think.