Generally, the Downtown Area is considered to be the safest, followed by the Heights and Journal Square. It also depends where you in the Heights, as it differs greatly within couple of blocks. There has been a lot of new developments around the east side by Congress Street. This is the location of the Elevator that brings you down to Hoboken's Light Rail. This area has nice views of NYC and provides a short and easy commute to the big city. I specialize in Jersey City and would gladly take you through the intricacies of the area and get you familiarized. This way have a better idea of where you'd like to be. Please feel free to reach out via email firstname.lastname@example.org or my cell 975 885 3961.
Wishing all the best,
Real Estate Specialist
273 Grove St
Jersey City, NJ 07302
973 885 3961 Direct
201 333 4443 Office
You can also visit general city data websites like city-data.com to get a feel for information about Jersey City as a whole. Otherwise while "scoping" the area out, start up a conversation with one of your "soon to be" neighbors. Due to the fact that deciding whether a neighborhood is "good/bad" or "safe/unsafe" is more or less subjective, you'll need to do some homework independently to see if it's an area you could see yourself living in.
For City Data visit: http://www.city-data.com/city/Jersey-City-New-Jersey.html
Also, seeking out an agent that predominantly works the Jersey City area, may also be beneficial. With today's tech. options, many agents have websites you can view at your convenience to find out if any agent has a niche market in Jersey City specifically.
Hope all works out and I've helped get you started with your search! If you have any real estate questions, please don't hesitate to contact me!
ERA Statewide Realty
Office: (908) 874-7797, ext 519