Tax Year Prior to 2020: What TurboTax version do I need - I live in NJ and work in NY?

You can use any version.  You just need to purchase two state software programs.

Will any version handle the credit for taxes paid in NY and apply them to NJ?

Since you are filing in multiple states and need to make sure the taxes and income are split properly, you should be using at least TurboTax Deluxe (you can get to it here: https://turbotax.intuit.com/personal-taxes/online/), BUT you should really go through what each version offers since you might have a more sophisticated tax situation besides multiple states and need a more robust version. When you click on the link above it will take you to the comparison page so you can see what version would meet your needs.