I live in New York. I will begin a job soon that is remote, but headquarters are in Delaware. Would I need to pay state taxes in both NY and DE?

It depends.  Delaware is one of a small handful of states that has a "Convenience of the Employer" law.  That means that if you work remotely for a Delaware employer for your own convenience, rather than because you were required to do so by your employer, then your remotely earned income is taxable by Delaware (as well as by your home state).

So, unless your employer is requiring you to work remotely, your income is taxable by both DE and NY.

In that circumstance, NY will grant you an "other state credit" for the tax you pay to DE on the dollars taxed by both states.  .  The credit in effect prevents double taxation, which is prohibited by a U.S. Supreme Court ruling.

If you have to file in both states, prepare your returns in this order in TurboTax:

1. Federal

2. Non-resident state

3. Resident state

Completing the returns in the correct sequence allows the program to correctly calculate and apply your "other state credit."