Can we rent a waterfront property in Cold Spring for a party?

If you mean a place where you can walk right down to the edge of the river - no.
You can rent Chalet on the Hudson in Cold Spring, which is across the street from the river on 9D, at the base of Breakneck Ridge. But Dutchess Manor is nicer, it's just a couple more minutes up 9D in Beacon - it's right on the river and they do weddings/parties/etc.