Do I have a claim in New York if unsafe conditions on someone else's property caused my injury?

Yes, property owners in New York are required to exercise due care to keep their premises safe for people who are on the premises. They're required to use reasonable care, and that means keeping dangerous conditions minimized. If there's snow or ice, to remove the snow or ice. If there broken steps or cracked sidewalk, to repair and maintain those conditions so that people will be safe when on the property.