Welcome!
Both Paramus and Hackensack are in Bergen County, one of the wealthiest counties in America, and a pretty damn fantastic place to live. There really isn't any way you can go wrong anywhere in Bergen County (which has A LOT of really small, one square mile towns). There are no bad neighborhoods, and from anywhere in Bergen County you'll be a short commute to Paramus or Hackensack through one of the many highways that pass through both towns.
Generally speaking, the towns from Paramus and north (ie Glen Rock, Ridgewood, Westwood, Saddle River, etc) are considered the more wealthy half of Bergen County. Quaint towns, good schools, expensive homes, etc. Sort of the Beverly Hills of NYC. It's not cheap to live here, but you may be able to find something in your price range.
I would probably focus on the lower half of Bergen County, which would be considered Hackensack and south (ie; Fair Lawn, Saddle Brook, Lodi, Rochelle Park, etc). More middle class, but still nice, safe and close to your jobs.
Your best bet for inexpensive rentals in the immediate area will be Hackensack itself (the closest thing Bergen County has to a "city"), Garfield, Teaneck, or Bogota. All are nice, safe towns, just more working class and affordable than the rest of Bergen County. You should be able to find 2BR for under $1,500 in all of these areas.
A step "up" from that would be towns like Fair Lawn, Saddle Brook, Rochelle Park, or Maywood. Middle class suburbia at it's finest. You should be able to find 2BR for under $1,800 in all of these towns.
As you go north of there, your prices will go up. Ridgewood is popular for it's downtown area, but it isn't cheap. Same goes for Glen Rock, and Paramus itself. You may be able to find something in your budget here.
If craig's list isn't doing it for you, check out Trulia.com if you want to work with a realtor, or use rent.com or apartments.com to find apartment complexes in these towns, then call them yourself to check availability.
Any more questions about any of these towns just ask! Just remember to factor in traffic, which you will be hitting traveling on Paramus and Hackensack's highways (80, 4, 17, etc). I would avoid Paterson and Passaic (west, across the river in Passaic County), but like I said, anywhere in Bergen County and you can't go wrong!
I've attached a map of Bergen County below, as well as a crime map (as you can tell, crime is virtually non-existent here) to have this all make a bit more sense. Best of luck!