I am currently filling a PR job for a client. The position description states that 70% of the accountabilities are writing related - writing press releases, web content, and speeches; as well as editing the written work of others. Based on these requirements, I have no choice but to summarily dismiss any applicants whose cover letters contain typos and grammatical mistakes. It's a shame because some of these applicants have neat backgrounds and would probably really enjoy this career opportunity.
I hate to see smart, skilled people lose out on great opportunities because of a "technicality." However, they've compromised their own candidacy by not demonstrating that they have the discipline to take the time to proof their own work. This sends a signal that they won't be a match for a position that requires them to write for the public, and to proofread the work of others.
Sometimes you can get away with errors in your writing when the job you are applying for doesn't emphasize writing skills, but that's simply not the case for most PR and Marketing jobs, and for many administrative and project management positions as well.
So if you're in a rush, or it's late at night, set your application aside and look at it again later with fresh eyes. You'll be surprised at some of the silly typos you catch in your own work. And who knows? You may just rise to the top as the candidate best suited for the next writing job that you apply for!