According to a brief scan of the comments on Yahoo! Answers, yes, a writer should probably have a website. I imagine all pre-Internet writers would have killed to have had such a user-friendly platform to promote their literary efforts. However, now that it's so easy to create a website (i.e. now that there's no excuse not to have one), you have to consider an entirely separate questions: not "Do I need a website?" but "Am I worthy (or shamless) enough to engage in self-promotion by having a website?"
Today, everybody writes. Literally everybody. If you made it through grade school, then you're a writer. The same is scarecely true for any other serious form of art (except for perhaps photography, speech, and culinary arts--or any other activity that we perform on a daily basis and that is somewhat connected to a serious art form). To keep it simple: not everyone paints, not everyone plays a musical instrument, not everyone sings, not everyone sculpts, etc... But everyone writes. Especailly now with text messaging: people who would never dream of creating a formal poem now regularly engage in crafting highly stylized, ultra-modern poem-esque works through texting, instant messaging, and emailing. Ay grl wutcha upta? O no u ddnt grl. Dont u b disrespktn daddy. ...Ami right?
So, if everyone is a writer, and if the comments on Yahoo! Answers are to be believed, then does it follow that everyone should have a website? I'll go ahead and spell out the obvious. Although everyone is a writer, not everyone takes writing seriously. Writing is a highly sophisticated craft. Before anyone in the publishing industry or in the reading public should worry about who is a good writer, they should first worry about who is a serious writer. Why? Because of the 10,000-hour rule. Nobody (with very few exceptions) who has ever made a worthwhile contribution to an art form has done so without first putting in at least 10,000 hours of practice. In the arts (and probably especially in the literary arts because stories and novels take so long to create!), hobbyists and fly-by-nighters won't stick around long enough to ever make a difference.
If you're asking yourself whether you need a website as a writer, you should first ask yourself if you are a serious writer. If you are, then you not only need a writer website, then you deserve one.