Chundles is right, the Cross has been cleaned up a lot. It's still gritty rather than pretty but I wouldn't hesitate to stay there - or to take my children either.
It's been my experience, after years and years of going there daily, that people will leave you alone, you won't be bothered by them unless you are involved in some way in their particular world. So, provided you're not a drug dealer or buyer you won't be involved in the seedy world of small-time drug crooks that support their own habits by dealing. And so on. And, as long as you take the same elementary precautions you would take anywhere, you're as safe in the Cross as you would be anywhere else. Perhaps even safer, as there are lots of police on the beat and usually lots of people around.
It's still a fascinating place, multi-layered. Gentrification is rapidly eroding the area from all sides. The whole of Macleay Street, from the fountain down to the harbour is now very upmarket, and the other side of the Cross, towards Rushcutters Bay, is going the same way very quickly. Some of the most expensive real estate is being built all around. Only one strip really remains relatively scuzzy, that's Darlinghurst Road from Bayswater Road down to Macleay Street. And that's so well lit and busy that you should have no problems there.
The Cross suffers more from it's reputation than from anything that actually goes on these days. As I said - provided you stay out of the way of the petty crims. They're definitely still around but they're not interested in you if you're not going to be making money for them.
The really good thing about the Cross is it's location. It is super close to everything. You can walk almost everywhere. If you decide you want a more salubrious location but still close to city and beaches, look at anything on the eastern suburbs rail line (Edgecliff, Bondi Junction) or almost as close but different (Darlinghurst, Pyrmont, Ultimo).
Good luck, and enjoy your 2 weeks