The quickest way to get the total NY 529 contributions from 1998 to 2017, would be to contact the plan administrator who manages the 529 plan. The contact information for the plan administrator should be on your 529 plan statement.
The NY 529 plan online account page only allows you to see contribution information back to 2003, and you have to select each year and add up the contributions.
[Edited 1.27.18 | 3:05 PM]
Does anyone know why Turbotax is asking for this information? Is this a new requirement for NYS? Can it be skipped? asking for 19 years of historical data is ridiculous - whomever is requiring it.
Easy peasy if you know where to look!
Go to the account on the 529 site/ click on the account
click on view details
scroll down to investments/ in that box down on the bottom you will see
Total Fund Assets
Principal This is the total contributions, what the IRS wants to know.
Told you it was easy.