From the Ben Gurion airport train station, a very short walk from arrivals, you can take a train to Haifa. A ticket may be purchased via a machine or at a booth, using cash or credit card. Take a taxi from the Hof HaCarmel station in Haifa to your destination. For more information, see the Israel Railways site. Note that trains and buses do not generally run on Saturdays/holidays in Israel, service stops on Friday afternoon and resumes Saturday evening.
You can take a direct Egged bus from the airport to Haifa. Take the 947 bus to the Hof HaCarmel station in Haifa and take a city bus or taxi from there to your destination. More inforamtion is availble on the Egged website. Credit card payment is not available on buses.
Sherut Taxi Service
Shared taxis, called Sherut, are available from the airport to all destinations in Israel. The taxi waits till it is full and only then leaves, so there may be a wait. There are 6-10 seats in such a car. The rate for a one way trip from the airport to Haifa is about 70 NIS (~USD$20). You may be able to use USD, but usually payment is cash only.
As you exit the terminal building, look for the supervised taxi stand. Give your destination to the dispatcher and you will be directed to the correct taxi. The fares from the airport are fixed, so the driver should not operate the meter. You may be able to use USD, or a credit card, but usually payment is cash only.
Avis, Budget, Europcar, Hertz, and other local companies have offices at the airport. A free public parking lot is available about five minutes’ walk north of Dan Carmel.