I am not a lawyer, but I am familiar with New York, having worked there. NY has some VERY strict labor laws, and while she would likely get into trouble, HR IS NOT YOUR FRIEND. By that, I mean that HR is there to protect the company, not to protect you.So, unless you have an iron-clad case for discrimination and have spoken to an employment attorney, going to HR could backfire.
Your best course of action is to tell the person who is doing this that you don't like your personal information being spread around the office. Tell her that it makes you "uncomfortable", and use that word. Then note it.
You want to keep a written record of everything because if this becomes a pattern, you want to be able to go to a lawyer, or to HR with a well documented pattern of harassment so that it's not just a "Well, she did this", but rather....
On February 13, 2017, at 2:15pm by the water cooler, I heard Jessica tell Robert "Can you believe she's 45! Wow, I can't believe she works in such a young field, she should find somewhere else to work" Then, later on March 6, at 11:00 AM, in a meeting about online widget marketing....
Make a paper trail and Document Everything That way, even if the behavior stops for a while, and then picks up again, you will not be caught without recourse.
But again, HR IS NOT YOUR FRIEND They won't automatically take your side just because you filed a complaint. The complaint needs to be solid, fully documented, and in the company's best interests to resolve in your favor. If you go in, be ready to supply your documentation and be aware of unforeseen circumstances such as being known as the person who filed against a coworker, which may affect your relationship with your other coworkers.
Try to resolve things with her first, then if things continue, get advice from a lawyer, then if the lawyer advises you, go to HR.