GarageBoy,
Not trying to be a wet blanket, but, without even getting into NYC ordinances, the term "dangerous knife" is used several times in Section 265 of the NYS Penal Code, but never explicitly defined. This seems to open up a large "grey area" for interpretation by the police and courts. If the DOB you list in your profile is correct you may want to consider this, at least for the next few months:
Section 265.05 Unlawful possession of weapons by persons under sixteen
It shall be unlawful for any person under the age of sixteen to possess... ...any dangerous knife;...
...A person who violates the provisions of this section shall be adjudged a juvenile delinquent.
Regarding the more general question on the legality of balisongs in NYC, I would ask the question, do any reputable gun/knife/sporting goods stores in NYC openly display balisongs for sale? If so, this would be a pretty good indication that, at the least, the sale and ownership of same is legal there although it would by no means guarantee that carrying one "readily available for use" on your person (as opposed to transporting it home from the store in a sealed package) would be allowed.