The proper place for biking is, of course, on a dedicated bike lane or bike path. However, when a bike lane is absent, the New Jersey law does not outlaw riding on the sidewalk.
"This is the sideWALK!" people hurl the words like a verbal projectile.
When my daughters were very young, I asked them to bike on the sidewalk on our way to town, while I would bike on the road. Pedestrians would smile at them and stand aside to let them pass. Because young children, on their small-wheeled bikes, wearing their huge helmets, are cute.When the children grow up and get to be bigger than their mom, pedestrians don't think them so cute any more.
But those nervous pedestrians have a point: The truth is that when you bike on the sidewalk, you are a menace. You and your bike are much bigger and heavier than the average pedestrian, and with any speed you have way more momentum.
So, while biking on the sidewalk is tolerated, it's not encouraged. If biking on the road makes you too nervous and you seek safety on the sidewalk, it's a good idea to act like a guest and show respect. When passing a pedestrian on a narrow sidewalk, have the courtesy to get off your bike and walk the few paces, or at least stop until they've walked past you. When there are children walking or playing on the sidewalk, please be extra careful.
Note: local laws on bike use of sidewalks may be different from New Jersey laws.
Princeton, for instance, does have a few sidewalks that are reserved for pedestrians, mostly clustered around the downtown area where the shops are. These sidewalks are marked with no-cycling signs. You can also find out which streets have pedestrian-only sidewalks by consulting the Princeton bike map, where they are marked by green dashed lines. A new batch will be printed soon; if you can't wait, you can check out a digital version of the bike map on the town's website.
For more information, the New Jersey Department of Transportation has a page on bicycling, with a link to the New Jersey Bicycling Manual.