Making airdrops with the brigades is rather tricky and takes careful coordination with an airbase and how you handle your transports.
Basically, airdrops early in the game are very difficult due to lack of good transport aircraft and what good transport aircraft you have, you need to do forced upgrades to get rid of the crummy bi-plane transports. After all this, you then need to concentrate most of your longer range transport assets into a single airbase.
From my experiments, you need to have something like 8 squadrons of transports on the same airbase stacked with the airborne unit. As I mentioned, the bi-plane transports don't get it done, so they all need to be the TB-3 transport or Li-2's. You have to make sure the airborne unit you are dropping has good moral AND a healthy ToE percentage. I think if a unit loses enough to go unready in the drop, it will probably fail.
Clearly it can be done since players have successfully pulled it off in some of the "cheese" manuvers to cut the rail lines a good distance from the front lines.
As far as the airborne brigades go early in the game, it is a tricky balancing act as others have mentioned a airborne brigade can slow down an enemy advance quite effectively. I have a tendency to yank them out of main line and use them as diggers/last line of defense. As the war goes along, the standard Russian rifle division doesn't exactly get better in terms of ToE/manpower. The Russian mountain/paratroop divisions are another story and keep a lot of "extra" in the ToE along with being much larger in terms of manpower over a standard rifle division. A paratroop division can easily hit 5 for a combat value in 1943 and its "elite" moral also means it can generally advance better than other rifle units.