Actually, there are tons of nice areas in Jersey City. And there are public schools downtown, and some of them are places I would send my kids. The Cordero School (PS#3 I believe) as well as the new middle school (PS #4) are decent schools. There are also several Charter Schools (OLC, Soaring Heights) and a Montessori school. When it comes to high schools, McNair Academic (also a public high school) is ranked #15 in the country for academics.
Sure, there are problems. It is a huge city. There are some developers who have taken advantage of the housing boom in the area--renovating or building quickly with their main concern being their bottom line, not the quality of the construction. There are also some great things on the market right now.
I currently rent downtown, and plan to buy in the near future. I also work downtown--I don't plan to leave Jersey City in the near future.
The waterfront developments are slated as luxury living, with light rail running through the streets and very high prices, and thus taxes. Just a little away from the waterfront there are some renovated condos with lower taxes (not new construction fares better). There are restaurants on Grove and Newark Aves, and PJ Ryans is a fabulous pub on Marin Blvd.
Like with any city, do your research! Don't move near the projects unless that is the type of neighborhood you want. Lock your car, lock your bike, but take your kids to the park. There are tons of little kids playing (with parents) in the parks (Van Vorst, Hamilton, and Liberty State Park are all downtown)
It's definitely worth a look!