Oita Airport is the closest airport, but you can travel to Beppu from Fukuoka Airport and Kitakyushu Airport.
http://www.ana.co.jp/asw/index.jsp
http://www.ana.co.jp/asw/index.jsp
http://www.ana.co.jp/asw/index.jsp
Oita Airport to Center of Beppu City (Beppu Kitahama) (About 1 hour)
(Transportation Fee: One-way 1 Adult 1,450 yen / Round-trip 2,500 yen)
http://www.oitakotsu.co.jp/timetable/airliner/air.php
http://rent.toyota.co.jp/top.asp
http://www.nipponrentacar.co.jp/
http://www.mazda-rentacar.co.jp/
http://www.e-budget.co.jp/yoyaku/
From Fukuoka Airport International Terminal to Center of Beppu City (About 2 hours) (Transportation Fee: 1 adult One-way 3,100 yen / Round-trip 5,500 yen)
http://www.rakubus.jp/rosen/fko_afk_bep.html
http://www.kitakyu-air.jp/access/tobus.html
http://www.jrkyushu-timetable.jp/cgi-bin/jr-k_time/tt_dep.cgi?c=28155
The closest station is JR Hito Main Line Beppu Station.
From Hakata Station JR Hito Main Line Express Sonic (About 2 hours)
From Ogura Station JR Hito Main Line Express Sonic (About 1 hour and 10 minutes)
From Kumamoto Station JR Hohi Line Express Kyushu Crossing Express (About 3 hours)
From Miyazaki JR Hito Main Line Express Chirin (About 3 hours)
http://www.jrkyushu.co.jp/train/
Station Rent Car
The closest port is Beppu International Tourism Port. There are ferries from Osaka and Shikoku Area. From Beppu International Tourism Port to Center of Beppu is about 10 minutes by car.
http://www.kanki.co.jp/page/kouro.htm
The closest Highway Interchange will be "Oita Trafic Road Beppu IC". From Beppu IC to Center of Beppu City will be about 20 minutes by car.