Not all writers do, it turns out. But you should. Okay, perhaps if your writing is a private, self-indulgence, then a web site is extraneous. For all other writers, consider it a necessity.
You can find lots of excellent references for how to establish an effective web site for your writing persona. Carrie Cuinn recently posted some great advice on her blog, with various examples for reference and illustration. Check it out.
In my case, my web site is due for another overhaul. I've been remiss accomplishing that because I built it using an old Microsoft tool (Publisher). My intent is not only to modernize my web site but also modernize my web development tool and methods while I'm at it. So it's on the To-Do list. Unfortunately it's keeping company with quite a number of other entries on that list.
Don't let my predicament deter you, though. There are tools that make simple web site development, well... simple. Many are free. Or you can hire out the work. I chose to take on the responsibility myself because I expected to evolve it. Often. With gusto. Also, HTML/XML coding comes in quite handy for creating e-books. But that's a completely different topic.