Welcome to USA! I live very close to NYC, 8 minutes away from the tunnels. I would say Hoboken is a very nice area with lots of bars, restaurants etc and i would recommend it for you since its mostly for young adults, single, etc.. , lots of residents here works in Manhattan , 1 bedroom, 1 bath you were looking at around $ 2500-$4000 depends. Weehawken is also a very desirable area a little cheaper than Hoboken, you are still very close to New York you can also look at the areas like Paulus Hook where it is a little far, the difference of 5 minutes from the areas i had mentioned earlier but it is very nice area with the view of Manhattan.
Where i live is the third safest area in Jersey City, if you need me to help you , you can email me at firstname.lastname@example.org take care and good luck to you and your wife.
Again, feel free to email me at email@example.com