Since you took your distribution in 2020, in early 2021 you will receive a Form 1099-R reporting the distribution and the taxes that were withheld.  You will include this on your 2020 tax return.  You do not need to send any extra checks to either the IRS or NY state specifically to cover the early distribution penalty.  It will be reconciled on your 2020 tax return, with no additional penalties.

However, you could consider sending an estimated payment to the IRS and/or to your state in order to pre-pay the additional taxes that you expect as a result of your early distribution.

As you are working through your 2019 tax return, TurboTax can help you calculate estimated payments.  There is a section in the Federal return and also in the NY state return.  

Here is an article to get you started for the Federal side.  How do I print estimated tax vouchers for my 2020 taxes?

As you go through your state, there will be a question toward the end of the process to ask if you want to make estimated payments.  Say Yes, and TurboTax will walk you through, including giving you information about where to mail the payments.