The CY in Jersey City is not exactly in a good neighborhood, but it should be safe. I don't think there have been any shootings in the immediate vicinity since the last time 50 Cent stayed in the Doubletree across the street.

And it's right next to the Pavonia/Newport PATH Station, which will get you into the city. See http://www.panynj.gov/path/index.html

To get to J.C. from EWR... either NJ Transit train or bus to Newark Penn Station. Then take the PATH train to Journal Square and switch to the other PATH train to Pavonia/Newport. Taxis from EWR to J.C. are very expensive, and the buses are very slow.

To get from J.C. to Bronxville.... Take the PATH to the World Trade Center, then walk a block or two east to the 4 and 5 subway station at Fulton St. Take the 4 or 5 train uptown to 42nd St Grand Central, where you can get the Metro North Train to Bronxville. According to http://www.slc.edu/visit, you'll need to take a taxi from Metro North to the SLC campus.

That's a lot of transit, though.

A better option might be to just rent a car and stay at the Residence Inn New Rochelle, which is only 5 miles from SLC (see http://www.slc.edu/index.php?pageID=2842). Of course if you're both under 25, renting a car is going to be expensive.