They need to have a grasp of basic grammar, yes. But do they need to know terms—the names of verb tenses, the names of the parts of speech, and so on?
This question was recently posed by a blogger at PR Builder, and I'm quoted in the resulting post.
It's an interesting question, and I think the answer is usually yes. Writing by instinct may often lead to great writing, but instinct isn't always reliable. (In the PR Builder post, writing by instinct is compared to playing music by ear. I think this is an interesting analogy, but most musicians earn their skill through study and talent. The same goes for writers.)
Grammar is the logic (as variable and capricious as it may be) of written language, and understanding that logic means knowing its terms. The rules of even simple things like comma placement require that a writer be able to say to himself or herself, "This is a subordinate clause" or "This phrase is nonrestrictive." Knowing how language works (and being able to make smart choices based on that knowledge) is, I think, what makes a writer a writer.
You may decide to break grammar rules, but you will break them much more effectively if you do so knowingly and with purpose.
And here's another, more self-serving reason: A professional writer will often come up against editors and copy editors who want to alter the writer's work. Often, the copy editors and editors are professionals who are making the right choices and helping the writer say what he or she means to say. But sometimes (rarely!), those copy editors and editors are less helpful and will damage a writer's prose. Either way, it pays to speak the language of editing, so you can communicate with them if it comes to a grammar "smack down."