I'd say any of the Paulus Hook or Liberty Harbor places fit your bill perfectly. Lenox, Quinn, Madox, Windsor, Portside Towers all come to mind specifically. Extremely quiet neighborhoods. Footsteps to the ferry, 5 minute walk to Grove Street, or if you're closer to the Liberty Harbor side, you can hop on the lightrail to Exchange Place too. Haus25 is very nice too off Columbus. Its very, very close to Grove Street station, but a bit on the pricey side. Basically you'd be good pretty much anywhere in downtown Jersey City east of Jersey Avenue. I would describe that part of JC as almost sleepy despite the density. One night I walked back from Hoboken to Liberty Harbor and could probably count the number of people I ran into on the street and this was a Saturday night. Everything closes early, its not a raging town like Hoboken is. The bars in Paulus Hook close at like 10:30.

At your budget you might even see if there's something available at 99 Hudson, which is basically across the street from Exchange Place.

I don't really think noise will be much of an issue unless you live in the building that has the Beerhaus on the ground floor. The light rail bell chimes occasionally but that's about it. Just be prepared that parking in the area is around $250 for people who live there, but I actually found a deal at the Montgomery Street surface lot for $180 if you're okay with not being in a garage.

The other area you might look at are some of the newer buildings in Newport especially closer to Hoboken. 75 Park Lane, 2 Shore Lane, The Beach at Newport. It is very, very quiet up there, you're a quick five-ish minute walk to the Hoboken PATH station and you get stunning views of Manhattan. Again the light rail runs through that area too and the mall and super markets are very close. You also don't have the traffic problems that you get in Paulus Hook where things get very congested especially on weekends and especially on Grand Street.