You always pay state (or district) income taxes based on where you live when you perform the work, not where the employer is based. There are 7 states that will also tax you if you live out of state but work remotely for an in-state company, but DC is not one of them.
So you will owe NY state taxes on all your world-wide income, since you are a NY state resident, regardless of where your employer is located.
However, if you perform any work in DC (such as, you are required to travel to DC for in-person meetings from time to time) you are liable for non-resident DC taxes on the proportion of income earned while living or staying in DC. You will need to review DC's rules on this, there could be a time limit, or it could be just one day staying in a hotel. When you file your tax return, you will start by preparing a DC non-resident return, and you will have to tell Turbotax how much income to allocate to DC. Then you prepare your resident NY return, which taxes all your income, but will give you a credit for DC taxes you pay on the same income.
You will want NY withholding, not DC withholding.
