It seems remarkable to me, in 2017, to even be discussing whether an author should have a website, but a quick Google search reveals ample evidence that it's a commonly discussed topic. And the consensus is by no means clearly in favour of websites as essential. "Go where the audience is", some sages will tell you, meaning Facebook, Twitter, Instagram and the rest.
I'm here to tell you that this is bunk. Websites are in no way old hat. You may not need a fax machine in the 21st century, and you can certainly manage without a photocopier - but a website should still be as important to a writer as their laptop, notebook and thesaurus (print or digital).