About 500 tons for their launching offshore platform. It is less than for a conventional wind turbine, but not much less. The force tilting the rig is measured from the bottom of the sea, not from the surface.
Indeed the launching way is very complex. And ships must avoid the unities during landing or takeoff, or go close to the rigs due to the low elevation angle (30-35° for a crosswind kite).
I do not think the current technologies could lead to an utility-scale AWES, even in 2030.
Only a giant (and infeasible!) unity like Dr. Beaujean's airborne wind turbine
(http://quak-equipment-projects.com/Visualisation/images/RechargeNews2012.pdf) could success in the end of the scale. In the other end of the scale, I see kiwee, but yet nothing between.