The closest airport is in Naples, approximately a 1½ hour. Then you can take the train or the bus to Salerno or Vietri sul Mare.
There are both ferry and a hydrofoil services departing from Beverello Pier in Naples and arriving at “Piazza della Concordia” in Salerno that is close the Railway Station and the bus terminal.
From there you can easily catch a train or a bus to Amalfi.
There is a bus service at the Salerno Railway Station that departs every hour is the bus of SITA .
If your coming from Rome there is also a bus service at the Tiburtina Station straight to Maiori.
For those coming from Amalfi there is the Marozzi bus service that services Amalfi, Maiori, Capodichino Airport and Napoli Centrale Railway Station.
Coming from the North take the A3 highway until you get to Vietri Sul Mare. Then take the coastal road called the 163 SS (Strada Statale) Costiera Amalfitana. Enjoy the view until you reach Maiori, then from Maiori’s seafront follow signs towards Amalfi.