How does one travel from Ben Gurion Airport to Allenby Bridge?

Up thru Jerusalem and down toward Jericho and the Jordan River.

Which transport would you suggest?

That a tricky question, since the Allenby Bridge is the border crossing between Israel and Jordan... I don't think you can cross to Jordan driving an Israeli (rental) car... Maybe someone else could help with this one....

Where are you going, once you get to the Allenby Bridge?
If Jordan is your destination, why not fly to Amman?

Otherwise as Mamamia mentions, don't believe you can drive an Israeli licensed vehicle into Jordan.

We are planning to travel from Ben Gurion, over Allenby Bridge, to Jordan. We will be met by a travel guide on the Jordanian side. I gather it's difficult to get across on one's own. Therefore looking for a practical way to cross. Maybe a bus? Please advise.

Take a taxi or sherut (shared taxi) from Ben Gurion to the King Hussein/Allenby Bridge (or taxi to Jerusalem, taxi to bridge, it's on the way.) On the Israeli side you will pay an exit fee (it was 130 shekels in December.) You go through passport control, and board a bus (a few dinars fee for baggage) and cross the river into Jordan. But you MUST have a Jordanian visa in order to cross into Jordan at the King Hussein Bridge!!! The King Hussein Bridge is the one crossing that does NOT issue Jordanian visas. (Send your passports to the Jordanian Embassy in DC.) You can meet your driver/guide there at the Jordanian immigration building that's just across the river. From there it's about 45 minutes to Amman. Allow plenty of time at the bridge, and check closing times, 3 pm on Fridays, for instance.
Mandi

Thank you soooo much Mandi!

I would strongly recommend anything but driving this yourself. We did just that and it was a disaster. We had all kinds of maps and still got lost in teh west bank...not where one wants to be lost. The logistics to get over were a pain as well and we ended up leaving our rental car, taking the bus, waiting...waiting...waiting...wondering if we were on teh right track etc. Take a bus or sherut from Ben Gurion as Mandi recommends.