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Try New Providence NJ. Based on a quick google search there are 1 BR rentals to be found under 2k. Meets your requirements 1 hour train ride, very safe and excellent schools.
1 hour commute is an interesting number. I used to commute to the City by train which was 1 hour ride. However, when you discuss logistics of it being door to door, it was more than 2 hours.
I do think you can have a 1 hour +/- commute (ride) really from anywhere in north/central NJ, and sometimes buses are quicker if you want to explore that option.
NJ is a big and diverse state, any specific type of demographics you are looking for? Age, race, neighborhood? I've been out of the rental market for a while, and know it has gotten expensive. $2k seems like its the bottom level for a 1 bedroom now, but I do think you can likely move into really any town if you look around and expand options.
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