Companies will build planes for whatever the market demands. We need regulatory agencies to impose carbon taxes on fuel usage so that inefficient planes are prices accordingly.
Aviation uses incredible amounts of fuel, and commercial planes are in the air more often than they are on the ground. My guess is that the embodied energy to operating energy ratio is lower for planes than for any other vehicle.
The beauty of a market-based solution is that you don't need to worry about details like that. If you price the externalities appropriately, the market will automatically optimize for the most cost-effective, environmentally friendly solution; grounding planes in situations where that makes sense and leaving them running where it doesn't.
And yes, a carbon tax would indeed result in "a greater cost of air travel". That's sort of the whole point.