Your extra curriculars look very impressive, you have a solid SAT score, and you have taken myriad AP classes which will not hurt you. So, depending on how extenuating your family's circumstance was, I feel that you may have a shot. However, I had the chance to ask an NYU admission counselour on what the "red flags" on an application are, and she did express that D's were unacceptable. However, given that you have a possibly valid reason behind it could help your chances. It does not take straight A's to get into NYU, and they are generally not too stubborn with C's (I got in with 2), but it is still a highly selective university. Good luck!!!