Manhattan neighborhood is a safe neighborhood but caution and safety is more important. It is very important to plan where you are going to, don't just wander around the neighborhood.
There's no place in Manhattan that is almost unsafe during the daylight. If you take a look at the crime statistics in New York, you will notice Manhattan North and South have the highest rates, but those are mostly grand larcenies and not street crime.
There are few places in Manhattan where there are not a lot of people around all time, so you're very safe. Back in the days, street grafters will try to take you into an alley to sell you a cheap tickets or something like that, but that is rare.
When you see many people in small area, many things are bound to happen from time to time. As long as you're at alert, you will be completely fine. If you walk like you know where you are going and you see a lot of people around, you're safe.
In daytime, you should feel safe in Manhattan but at night it is important to stay on the main streets where there are other people walking. There has been huge progress in safety in the city. There are especially negative periods, and a higher rate of dangerous crime during the crack epidemic, but that has subsided.
People visiting Manhattan have always remark about how they were fearful but were really surprised by the safety, accessibility and friendliness. There are crimes that have happened in Manhattan in the past. On July 23, 1991 the body of a four-year old girl was found in a cooler on the Henry Hudson Parkway in center Manhattan.
The identity of the child, dubbed "Baby Hope," was unknown until October 2013, when a man named Conerdo Juarez is arrested after confessing to killing the girl, his cousin Anjelica Castillo, and dumping her body in the cooler. On July 4, 1996 a 22 year old drifter John Royster brutally beats a 32-year old female piano teacher in central park, the first in a series attacks over a period of eight days.
Royster would go on to brutally beat another woman in Manhattan; he raped a woman in Yonkers and beat her to death at Park Avenue on the Upper East Side of Manhattan. He was later sentenced to life in prison without parole in 1998.