Bedminster is a beautiful town. I work in an office building in The Hills, which is a large master planned subdivision built in the late 80's/early 90's. There's tons of homes in there ranging from condos to $million + homes. It is an expensive town, but the schools are very good and it's very safe.
The drive to Basking Ridge is 4-8 miles up 287 depending on where you are, and it'd probably be 15-20 minutes during rush hour. Jersey City is around 30 miles east on 78, and could be easily over an hour during rush hour. There's a train station a few miles away in Far Hills, but its small, and I think you'd have tot take the train to Hoboken and then the PATH to JC.