The lorry drivers do have a separate cabin that they travel in. The cars are in completely enclosed double-decker carriages, but the lorries are open to the elements.
Less open to the elements now.[1] The lorries now ride Eurotunnel in cars with a solid roof and steel truss sides. The roof is to prevent projecting objects such as antennas from contacting the overhead power line (that's happened) and the open sides are to allow fire extinguishing (that's happened, more than once.)
Such are the practical problems of a "rolling highway" Eurotunnel has had to solve.