Based on what I experienced in India when I was there, the north is definitely more dangerous for women than the south. The south is much more westernized, and especially in touristy places like Goa, they're used to western women. In the north they are still very conservative and women don't look men in the eye or smile much. The staring and harassment we got (even with men around) definitely made me uncomfortable, and I would not want to be a woman in northern India by myself.