Yes, absolutely. Downstate Correctional Facility is a New York State prison and accepts mail including postcards through standard USPS delivery. There is no issue sending directly to his new location.
If you had already sent a postcard to the county jail before the transfer and it did not reach him because he moved, InmateAid will resend it to Downstate at no charge. Just email aid@inmateaid.com with the details of the transfer, his new facility information, and the team will take care of getting it back out to the right address.
Downstate is a reception and classification facility, which means your son is likely going through the intake and assessment process that determines where he will be permanently housed within the New York State system. That process typically takes several weeks. During that time mail can move more slowly than it would at a permanent facility because of the processing that is happening simultaneously, but it does get there.
One thing worth knowing for the immediate term is that his permanent facility assignment may change after classification is complete. If he gets transferred again from Downstate to a permanent facility, let InmateAid know and any mail caught in that move will be resent at no cost as well. That policy applies regardless of how many transfers happen within the system.
Keep sending. Staying connected during the classification period matters, and a postcard arriving during that stretch tells him someone on the outside is tracking where he is and thinking about him.