How important is it for a writer to be fast when using a keyboard?
In my case, the answer was: not very important at all. I'm the author of 40 books and hundreds of magazine articles. I've made a living as a writer since about 1990. I never learned to touch type well; my typing speed is about 30 wpm now. As my writing needs and output increased, I turned to voice recognition using Dragon NaturallySpeaking, which also saved me from a nasty bout of RSI.
Almost anyone can talk way faster than they can type. Dragon is not a perfect solution by any means, but it's really good once you learn its quirks. You must learn to use it effectively, and it's way better on Windows than on the Mac (ironically, for me, since I write mostly about the Mac and Apple products).
I use Dragon to compose first drafts of book and article manuscripts. I usually then edit by typing. Things like email I usually type, unless I know in advance that it's going to be a long message, in which case I'll shift to voice recognition. I also tend to write short pieces like this answer by typing.
See Dean Wesley Smith's recent article about writing speed: http://www.deanwesleysmith.com/?p=3204
As Smith says, "I don’t type faster with my little four-finger typing, I just write more hours than most... I am considered a fast writer because I spend more hours writing. Nothing more."
The answer depends upon what type of work you are getting into.
Just because you can type fast, doesn't mean you can write well. But writing fast does have the added benefits of allowing you to quickly lay down a rough draft, tweak the hell out of it, clean it up during editing, and publish the final piece. Also, if you are going to be writing for a fast-paced publication that requires you to submit a certain amount of articles per day, you would help yourself by being able to type faster.
Seeing as I write for both (I've previously written for Mashable, CMSWire, and others), I have managed to find a good blend. I can usually write 700 - 1,000 words in a rough draft in the span of 30 or so minutes (assuming I don't need to break out of my flow to do research/fact checking). But it could require an hour or two of editing and fact checking to make this blob of text into something presentable.
For more long-form content like editorials and essays, however, I usually take my time and pay more attention to the flow of the article. It is especially important because you have to keep the reader engaged and captivate his/her attention for a longer timespan (which is critical for the plentiful amount of readers that have ADHD in my industry).
In the latter case, writing speed isn't as important. Creating a story that people will want to read is critical, though, and even when I can pump out an impressive 125+ words per minute, little of that helps with the basics of crafting a well-written article. It might help get ideas out of my head faster, giving me more time to rearrange things and produce a better product, but that can only take you but so far.
But you are a writer, right? You are always looking for ways to improve your writing. Improving your writing speed can improve your writing. So take it slow for now; build up your writing speed over time.
TL;DR: Writing speed helps immensely in certain situations, but crafting a well-written product, in the end, requires creativity and dedication — something writing speed can't compensate for.
I don't think that FAST typing is important, but I do think it's important to know how to type. Not to be able to touch-type is a handicap, not just in writing but in the job market overall. With the internet it's easy to learn to type or get better at typing--there are LOTS of sites for typing help.
Typing qualifies as a communication skill; writing is all about communication; why not have as many skills at your command as possible? Even if your typing aint great, it is better than not being able to type at all.
But as far as whether being a fast typist makes one a better writer--no. Not in my opinion.
Think about how fast you need to type in order to hit your word count goals: if you can write a thousand words in an hour, then you'll have no trouble writing things quickly (only fifty hours to complete NaNoWriMo, for example).
But a thousand words an hour is less than seventeen words a minute - you don't need touch typing to reach that pace.
... (this may be begging a down-vote, but...) Irrelevant. For the majority of history we've had writers who didn't have access to keyboards. I'm loath to think that the great writers from two-hundred years ago were worse writers because of this. They may have been more prolific had they been able to write faster, but that has nothing to do with the quality of their work.