Did you try emailing sales.tlv@thy.com? I've found them very helpful with other issues in the past. I don't think there is any other airport outside of Turkey that has as many TK flights as TLV does!
In general IME you get much better results from local ticketing offices of Turkish Airlines than you do from the toll free call center.
I emailed the NY office a few days ago, this was their response:
Basinet seats are payment seats. In order to confirm a basinet seat JFK IST it will cost USD 79.00 per person.
Or
If you do not want to pay, you can request basinet day of flight if it is avl check in agent will assign.
Best regards,
Fatima ABDUL SHAKOOR- New York Ticketing