This is a very timely and potentially tricky issues. Judging by the comment posted to the linked article, NYCB's decision to pursue such a policy may be linked to twitters by a company member.
I have mixed feeling about this kind of policy. Certainly, even as far back as the early days of social media (MySpace primarily), I've had concerns about the content some dancers did and still do post online. For instance, bad-mouthing artistic directors or posting questionable photos on publicly viewable websites.
I think some of the issues reflect the fact that professional dancers trend begin their careers at a very young age, and the ballet-student world can be very isolating. So they don't have the time - or freedom - to make the mistakes that "civilians" can get away with in high school or college years. But they often have the income to buy cellphones/computers long before their peers.
So, I think that a little lesson in thinking before you type or post would go a long way in the ballet world. You may hate x's guts, but it's a small world and who knows when x might come back into your life as a competition judge, choreographer, new AD, company teacher etc. Or that drunken pic you posted may seem innocent, but could come back to haunt you because if someone on the company staff notices it you could have a lot of explaining to do in terms of missed classes, or (for those under the drinking age) a breach of contract for violating the law.
That said, a company like NYCB has to tread very carefully to avoid being seen as a censor. As would be suggested, does a company have a right to prohibit a dancer from mentioning another dancer's injury - so long as the injured dancers gives permission. What if the dancers are husband and wife/civil partners? Siblings? Is it better to try and keep the not-so-nice stuff hidden, or will it blow up in your face?
Do you risk taking the spontaneity and uniqueness out of dancers' blogs or twitter postings. I think the Royal Danish Ballet had a story or two that didn't make them look good, but have moved well beyond that by all but encouraging dancers blogs, and opening the theatre up for at least two very honest documentaries (one shown, one soon to be shown).
Perhaps a middle ground could be found by talking each year to all the dancers about social media, and the dangers it can hold. Remind them that there is little that is truly private on the web, and to think before you post pictures or text. Give some examples of what can happen, and what is and is not wise in terms of internet presence. Also, it might be better to encourage dancers to be open about their blogs and twitter feeds so that if something not quite right is posted, the company can chat with the dancer. Make it a learning experience.