airport to cinque terre

Milan, Genoa, Pisa or Florence.
Whatever is cheapest and most direct from where you are! :O)

You are better off choosing the airport that works best within your flight time options and/or budget as Kathy states.
Pisa or Genoa are the closest and best options but not close enough that it is worth flying in multiple hours later than say reaching Milan much sooner.

Florence is also a small airport and in between distance/time wise, it could also be an option but since it seems even less flights go into here than Pisa or Genoa is unlikely going to be your best option.
Bologna could work too but would only choose that if times were much better than others which is also unlikely.

Monterosso is approximately half way between Pisa (PSA) and Genoa (GOA) airports, the closest airports to the Cinque Terre.

From either of those airports, you'd need to take a short bus shuttle ride to the main station (PIsa Centrale or Genova Piazza Principe) and take a train from there to Monterosso. The train ride is just over one hour from either place.

I don't know where you live, but I'm assuming you are in North America, since the majority of posters here are.
PSA has one non stop flight from/to JFK (Delta) in summer and also several flights to the major European hubs.
GOA has no direct non stop air links to North America, but has flights to other European hubs.
Milan (MXP, or LIN) is over 3 hours away by train, but it has numerous flights to everywhere, including North America (from/to MXP).
Florence (FLR) is less than 3 hours away, but with no non stop links to North America.

I usually fly out of Cinque Terre to USA from Pisa, spending the night in Lucca.