Yes, if the system is working properly, the conviction is still on there.
You could talk to a lawyer and find out about getting it expunged, which would effectively "remove it" from your record. This is typically expensive and not even an option in some states (not sure about NY).
Assuming you don't have any other major problems on your record, I would hope that a DUI that is nearly 30 years old wouldn't cause too much of a problem for you, but you never know.