Family of 5 moving to NYC

I’m moving to NYC with my husband and 3 kids For a new job. I need help with finding a safe neighborhood and decent rent. I’ll be working off Park avenue S, Flatiron district. I’m willing to pay up to 3k for rent. Please help with recommendations. I’m open to Jersey locations or Westchester- anything that will give me more than 1 bedroom at a decent price and is safe.

Edit: my kids are homeschooled so school/daycare is not a consideration. i’ll be working hybrid and there’s a parking garage at work if I need to bring my car and drive to work for the few days in office I’ll go in.

based on the comments it looks like I need to go up to at least 4k.