How much do NYC rental prices decrease during winter? On say a 3k/month rental right now, how much cheaper would it be in January?

Apartment rent doesn't change by the season. If you are referring to hotels or other short-term rentals, then you will likely find cheaper rates in the winter, although I don't know by how much. Those kinds of things vary considerably.

It would still be $3000

The rents don't get cheaper. For the rent controlled apartments when there is a vacancy the landlord is allowed to raise the rent by certain percentage set by the Rent Guidelines Board. The key word here is "raise."

For a non-rent controlled apartments the rent increases may vary due to market forces, but these forces have nothing to do with the seasons. But even then we talking about rent increases, not rent reductions.

NYC long term rentals do not have a seasonal bias. You might get a better deal on a one year lease if you move at the beginning of August as people generally like to start new jobs etc in September

However, if you meant to ask about short term (tourist) rentals, cheapest in Jan-March when the weather is worst.