<p>When you use the PHP <a href="http://us2.php.net/manual/en/function.copy.php" rel="noreferrer">copy</a> function, the operation blindly copies over the destination file, even if it already exists.  How do you copy a file safely, only performing the copy if there is no existing file?</p>

I think you answered your own question - check to make sure the destination file exists before performing the copy. If the file exists, skip the copy.
Update: I see you really did answer your own question. You mention race conditions, but if you do find that the file already exists, how do you know that:
the file that is already there is really the one you want to copy
the other process copying the file has completed its job (the file data is all there)
the other process copying the file isn't going to fail (and leave an incomplete file, or delete the new file)
I think you should consider these questions when designing a solution to your problem.