A-ha! Finally, I can be of use to someone here...
I have done this. It's a long way to Tippera... oops, to Manhattan from Islip.
By public transportation: at the airport, catch a taxi from Islip to the Ronkonkoma stop on the LIRR, which would be about $4.00. The LIRR into the city from Ronkonkoma is about $12.00, I think - check
www.mta.nyc.ny.us
By taxi: it's quite expensive, like $100. Ask the cab/car driver before getting in.