I've had this in the past and eventually gave up. There's some sort of known issue with getting a Mac to print to a printer that's shared by Windows. I found plenty of people having the same problem, and nobody actually fixing it.
That's kind of what I did. I didn't consider it an important enough problem to spend too much time on. One of the assistant headteacher's did moan but there's a computer in her office that the deputy head uses so I told her to use that instead. She moaned enough to the boss that I spent an hour a week last Wednesday but to no avail so at least I could say I've tried.
I then decided to try plugging my Macbook into the printer via the USB cable connection. It worked. So I then emailed my boss and the two Macbook users to say USB printing works. Predictably the boss wanted a demo so got his Macbook out. It failed.
We then had another go, I downloaded the drivers for the printer, added the printer and it worked wirelessly. I replicated this on my boss' Macbook and it worked for him as well.
I sorted the other Macbooks out as well but not before the assistant head's Macbook replicated the authentication issue. I realised she was using the account copied over from her old Macbook (now mine) to her new Macbook Air. I told her to use her new account on the Macbook and it worked fine.
It doesn't help that the printer is on the admin network, for office staff and senior teachers to use, but the wireless points all connect to the curriculum network so trying to get the Macbooks to see a printer on the admin network through the curriculum network wireless connection is very painful and takes several minutes to show the admin server so I can then choose the printer.
I did try to explain to my colleague that it was a difficult problem, with no obvious solution to be found online and my experience/expertise lies in Windows and not Macs. She asked for a Macbook and so I wasn't exactly 100% sympathetic to her problems. Had she accepted a new Windows laptop she'd have been printing wireless from the day I set her laptop up.