I suppose I have some opinion to share about this. I have an M.F.A. in Creative Writing and currently work for an academic publisher--not quite what most people think when they talk about "publishing," but largely the reality for most people in the industry (academic and small press is a lot more accessible). My undergraduate degree was probably pretty close to yours (I have a B.A. in "Language, Writing & Rhetoric" with a minor in Digital Media) so I was definitely in this boat at one point, but my path is very different and shouldn't necessarily be seen as the one for you.
Do you need an advanced degree to get a job? No. Absolutely not, but certainly it depends on what job you're trying to get. If you want to be a publishing agent at a major literary house in NYC--well you've got a long way to go, and no amount of degrees is going to open that job realm up to you. So much about the publishing industry is networking. Honestly, so much about ANY industry is networking. When I was applying to jobs out of undergrad, it was 40+ applications a week with maybe one interview every now and then. It was miserable. I eventually learned some common sense about job hunting--started using LinkedIn to network with human resources in the company I wanted to work for. Other tactics are much more straightforward, like walking in the front door and asking to speak to a hiring manager. Having a face to the name is an old saying but I think it's been useful for me.
Anyway, I personally would continue to think about other options than a Masters in Publishing from NYU. I have a couple friends that have gone to NYU, and the only reason it was a great experience for them is because they come from wealthy families. Sorry to say, but there's a reason why NYU is the "Number one dream school in America"--because for most people it is nothing short of a pipe dream. Their tuition is through the roof, and the costs associated with living in NYC are staggering. The additional catch with NYU is that they offer VERY LITTLE in the way of financial aid for M.A. programs (M.F.A. too, yikes). For me when I was applying to MFA programs, I couldn't come to consider NYU, no matter how much my writer-ly sensibilities make me want to live there. They offer some scholarships but they're very hard to get, almost always based on merit (there's always someone better than you applying to NYU... well that was my thought), and they don't really dent your costs. Yeah you could take out loans, and sure that's a personal decision, but for me it was an easy thing to push aside.
Now consider this: you don't need to get a degree in Publishing to become a publisher. Getting that kind of job is going to be a task beyond what your degree says after a couple years in grad school, and a Masters in Publishing doesn't really qualify you more than the guy with a degree in Digital Rhetoric & Writing, Digital Media, Print Media, Print Publishing, or really any M.A. in an English, Rhetoric, or Journalism field.
All I'm saying is, there are great programs elsewhere that don't require you to go into 50k+ debt (and you will). There are even competitive M.A. and M.F.A. programs (almost entirely NOT in NYC) that will pay you to attend. For me, I went the MFA route and got paid a stipend on top of tuition + fees + medical benefits. But again, that's not really an option for everyone (if you're not a fiction writer or poet, you're not going to be by the end of undergrad), and an MFA is by no means a professional degree. In fact that's something to keep in mind... a lot of M.A. degrees are professional, but fine arts never really is.
Anyway, look into your options. If I might make a suggestion, look at Digital Media, Rhetoric, Writing types of degrees (I'm thinking about another M.A. in Digital Media) both here and abroad. NYC is a big dream for us types, but do you realize that for less money you could get a comparable degree in Edinburgh, Prague, Oxford, London, Paris, et cetera insert-dream-European-city? My life has been GREATLY enriched by doing study abroad as an undergrad, and I heavily considered an M.A. in the UK for a while. Just saying, it's pretty neat.
Erh, yeah, sorry, 3a.m. rambling.