I love Uber. We use it frequently when traveling. I have used it in large cities at night (Boston, Orlando, New York) and never felt unsafe. One thing I will do though is forward the info to my husband.
That way he has the driver info, license plate and description of the car. Because we travel a lot in cities we're not necessarily familiar with, I always let at least one other person know my plans. That way if there's an emergency (car accident or health emergency) at least they have an idea of the area I was in and what my plans were if they need to start looking for me.
I've worked in law enforcement environments and have dear friends and family that are in that line of work. Their mantra is be aware of your surroundings. If something doesn't look or feel right trust your instinct. When I was single I didn't think twice to ask security to walk me to my car if I was uncomfortable. I would drive by my house to make sure everything looked in order before pulling into the driveway.
Bottom line...if it makes you uncomfortable, don't do it. New York has multiple traveling options. Be safe and enjoy being there. We love visiting there. 😊