The threat of airborne in a large battle (main army attacks, such as D-Day and the never-happened 1985 Soviet attack) keep people scattered about in a variety of support positions far to the rear.
In the Pacific of WW II (official records show no jumps; I know people who made them), platoon-sized drops into difficult supply lines, such as New Zealand mountains, caused great distress in specific cases. There seem to have been 6-12 of them. My sources were in their 80's and living in a VA hospital.
Currently, however, isolated jumps into Columbia, particularly by special forces, have eliminated several drug cartel holdings and at least one cartel. Airborne insertion of small units still is practical in small numbers.
On a large scale, airborne seems to be about dead. The 82nd has been discussed for elimination repeatedly; 101st isn't airborne any more, but air mobile. 17th airborne corps exists to support three airborne divisions at Ft. Bragg, but no one seems to think that likely. The barracks to support those men are not existant, any more.
My Marine Corps has a few small units for airborne insertion (force recon, specifically), but uses them sparsely in that role. Air insertion by chopper is much more common.
I'm told by some friends who know there are many more drops than reported, but they are a squad to a platoon on clandestine missions (such as Columbia). Deniable missions which 'never happened'. Even Time Magazine, a few years ago, cited 26 insertions in Columbia on the front cover.
There's a Marine effort to use drones to put a squad of Marines each anywhere in the world in two hours per drone. Can't imagine who thought that one up, but (hopefully) they know more than I do. The cost will be incredible!
All-in-all, the airborne has some role today, but not in divisional drops vs 1st-world military forces. Small insertions are likely to stay a very long time (watch out for drones on the street down from your lane)!
"Some people live an entire lifetime and wonder if they have made a difference in the world. Marines don't have that problem."
President Ronald Reagan, 1985