As someone who has lived in the Heights for 20 years, it's very chill here. not a ton of bars or anything fun really, but not super far from downtown where there is a ton of food/drink/etc. We have never felt anxious around here and I used to work super late and come home walking by myself at 2-3am routinely with 20lbs of camera gear in my bag and thankfully it's been incident free. I don't do the late nights in recent years, and use Uber a lot more... but overall we are pretty damn content here and were lucky enough to find our most recent rental, and the owner lives next door so we don't have any challenges dealing with absentee landlords etc.

I should have given a little more info about Jersey City Heights-- it was its own city until the late 1800s, and it is pretty geographically separated from the rest of JC--it's up on a cliff overlooking Hoboken. It's easy to get to journal square or Hoboken. I love Central Ave, which is the local main street. It is extremely walkable with many small businesses and family owned restaurants. We are moving out of a beautiful old townhouse that is divided into two units. I personally prefer the older buildings to the new "luxury" construction. The basement one br unit in our building was rented for 1200/month, so I think you should be able to find a nicer, non-basement 1br within your price range.