I've shipped cars using an open rig, including my 1937 Packard, and never had a problem. Most new cars are shipped this way (although still swathed in their factory protection).

I can understand those who want to employ a closed trailer for shipping a rare antique or exotic or show car, and the higher cost can be worth it for cars rarely driven in the elements.

But if you expect to drive your car in the wind, rain and snow then it's not going to suffer any worse on an open rig than it would if you drove it yourself -- and likely alot less. I'd request to have the car placed on the upper portion of the rig. Those on the bottom portion could be subject to oil/fluid drips from the cars above.