My short answer… YES!!!
My longer answer…
It's definitely important, however, more from the inverse perspective. Here's what I mean:
Good writing is rarely noticed or rewarded. In most business or personal situations, people generally don't notice if something has been written well. It's expected, and nothing in particular stands out to them. However, it's when someone has a spelling mistake, poor grammar, bad punctuation, or worse… texting jargon that readers notice… and most often think less of the writer.
It's not insignificant. At times, I have people tell me that it's petty for someone to judge them because of spelling errors or grammar. It is true that people sometimes make sweeping generalizations about someone because of some relatively minor mistakes. However, it's not invalid to to make some assumptions based on how a person communicates in writing. Poor writing often is indicative of carelessness, a misjudgment of what's appropriate, being under-educated, sloppiness, or other negative traits.
It can be a "CLA". When someone writes poorly, it can be a Career Limiting Activity. It's often difficult to justify promoting someone into greater responsibility or scope of influence if his writing might represent the department or company poorly. A manager may lose credibility and respect from subordinates if his written communication to them is full of errors. Similarly, he may lose respect and credibility from his own superiors as well. It happens, but it's rare to find a senior level person in most any organization with poor writing ability. How one communicates in writing carries a great deal of weight in their career advancement potential.
So what should you do? Be careful, practice, double-check, and get help. All of us regularly make typos, mis-spell words, make awkward grammar choices, and make other mistakes (I occasionally get emails from people correcting my articles). I proofread everything before I publish it, and still find errors days and weeks later. Similarly, I always reread emails before I click SEND. Yet, at times, still see a typo when someone replies and includes my original message. However, when writing in a business context, it's critical that errors are a rare exception rather than the norm. Would recipients be surprised to see a mistake in your email, or would they be surprised when they don't find one?
When it comes to resumes, cover letters, thank you notes, emails, or any other written communication in your job search… there is no excuse for any kind of error at all. Communications in your job search are expected to represent your very best. An employer generally expects that a candidate will present their best effort when trying to make a positive impression. Writing errors when they are presenting their "best" creates an impression that their "average" work will be sub-par.
Does writing ability matter? You bet! Put in the effort to improve your writing, and you're likely to improve your chances at getting a job, and advancing your career.