Working remotely for an NY company does not always subject you to NY state income tax, there are several factors that must be weighed, see this memo from NY.
https://www.tax.ny.gov/pdf/memos/income/m06_5i.pdf
Assuming you do owe NY tax, you would file a non-resident NY return and a DC resident return. You will owe NY tax on NY-source income (but not other income like investments) and you will owe DC tax on all your world-wide income. DC will give you a credit for tax you pay to NY.
You should definitely have NY tax withheld. For most incomes, DC tax is 1%-2% higher than NY's income tax. If you have full withholding for DC, you will have a lot of extra tax withheld and get a huge refund while not having the money in your pocket. I think it would work to have NY tax withheld, no DC tax, and then make estimated payments to DC on their web site, in the amount of 1.5% of your income. (This is a very quick off the top of my head guesstimate.) You can make the payments weekly or monthly but no less often than quarterly.
