LaGuardia International Airport (LGA), John F. Kennedy International Airport (JFK), Newark Liberty International Airport (EWR)
Given New York's great public transportation system you can essentially fly into any of New York's airport and make your way into the city. LaGuardia is technically the closest airport as it is located in Queens and not far from Randall's Island, but really any of the airports will do. If you do fly into LaGuardia you can take a shuttle into Manhattan or Uber's can be accessed from outside the baggage airport.
The Uber price will vary widely depending on where you go, as will a shuttle. Generally an Uber to Manhattan from LaGuardia or JFK will be far cheaper than a cab, so it is our recommended method. An Uber from JFK to Manhattan will be about $40 and from LGA to Manhattan will be about $30.
Generally the cheapest time to book flights for Governors Ball will be about six weeks before Governors Ball. This means you should be focused on booking your flight in April 20, 2018. This may seem late to you planners out there, but statistically it's the cheapest time to book flights. That said, if this is too late for you book it earlier. One final note is that plane tickets for Governors Ball tend to skyrocket about two weeks out, so certainly don't wait that long to book a Governors Ball flight.