Manhattan is very expensive to live and $1500 is a small budget. With that said, Inwood and Washington Heights (above 165th Street on the Columbia Medical School campus) are both very safe places to live and have spacious one bedrooms and studios at or under $1500. The Uper West Side is very safe but likely out of budget.
You might want to check out other bouroughs as they can be cheaper then Manhattan and provide more living space for the cost of rent.
Also, check area information section of HotPads NY page.
*This was voted the best answer from our community of users, not an answer, opinion or advice from HotPads.