You are talking about the line for "Seller's Signature", right?
That is correct.
What would I have to do if I was the new owner?
You will need proof of the purchase price for sales tax purposes -- have the seller fill out the appropriate NYS vehicle sales tax form, most often the "Affidavit of Sale/Statement of Transaction" (DTF-802) available at any motor vehicle office, by request from a DMV Call Center or from the DMV Internet Office.