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Tribeca near Canal doesn't really favor PATH trains over other lines as there are the A/C/E, 1/2/3, and R running through the area and N/Q, 4/5/6 and J/M in the adjacent chinatown and city hall areas.
I think since the kids are small, you can probably swing a decent place with that budget in Brooklyn Heights which would give you a very short commute to Tribeca and would put biking and taking a cab as fairly easy and inexpensive options when you feel like doing that instead. The neighborhoods pretty family-oriented with quiet, tree-lined streets with rowhouses and lots of nearby parks. A bit further out are the previously mentioned Park Slope and Prospect Heights. Probably take those and the other mentions and then go on Trulia or Streeteasy or similar and see what you get for the different neighborhoods mentioned and then try running google map searches for the distance to your office (you can just drag one of the pins around and see how long transit takes).
Why would anyone deliberately move to the cesspool that is NYC?
Per the OPs post
"Right now, I'm just trying to gauge if the move would be worth it as the salary is a nice bump from what I'm currently making and NYC could open many opportunities for my future career."