Could anyone recommend a cheap place to live closest to NYC?

Hi Chantelle - upper west side (or "UWS" as you'll probably see it in real estate listings) is a good location to live, but it can be expensive. The smaller space you're willing to live in, the cheaper it'll be. If safety is your main concern, I would advise you to find an apartment in a doorman building. You'll get less space for your buck, but a doorman bldg typically has more of a luxury feel and definitely provides more safety. There are lots of other locations in Manhattan that are nice as well. Generally, I stay within these parameters when I'm searching for an apartment:
Anywhere between 14th Street & 96th Street, 1st Ave & 7th Ave. If you go below 14th street you'll want to stay on the west side. The east side is "Alphabet City" and, as adorable and kindergarten-ish as it sounds, you don't want to be there! If you stick with UWS you can probably go as high as 110th Street and it'll still be pretty safe.
There are great areas outside of Manhattan that are also nice and safe and less expensive. You just have to know where to go. Certain parts of Queens (Astoria, Sunnyside, Long Island City) can be nice. Brooklyn also has a bunch of good areas but (Williamsburg ... NOT EAST WILLIAMSBURG, Greenpoint, Bay Ridge, Brooklyn Heights).
I would be careful in Brooklyn though. There are parts that are really nice, but walk a block or 2 in the wrong direction and it gets creepy.
All in all, NYC is a great place to live... you just have to be mindful of the nice areas and not nice areas and stay in the nice ones. I'd say there are more nice areas than bad ones.

Hope this helps! Good luck :)