Yes you can damage an enemy in the air they are not invincible or invulnerable while in the air depending on the cause of whatever placed them into the air.
Both Tiny, Rubick, Tusk and multiple heroes having enemy in the air using a skill and you can still attack them unless they are outside the attack range of the hero or the hero has such a slow attack speed that the normal attack cannot connect to the unit while in the air.
Tiny using toss allows him to use his avalanche on the tossed unit or hero increasing the damage done to the unit or hero.
Stuns like the one from Sandking, Lion, Nyx Assassin will also throw the enemy unit into the air and you can still attack them.
Also there is this skill of Earthshaker to jump when the hero has an aghanim's scepter and uses the Enchant totem skill and Earthshaker could still be attacked in the air. (It's funny to see Earthshaker land in a heap when he dies in the air)
Here are some of the skills that will make you invulnerable when you are in the air (damage is only from the skill itself, no way to damage the hero while under the affect of any of those spells):
- Invoker's Tornado -> damage after you land but none while in the air.
Eul's Cyclone -> no damage if you're the owner, but deals 50 damage if it's used by an enemy, this means that Eul can stop enemy blink daggers.
[Cyclone]: A target unit is swept up in a cyclone and made invulnerable for 2.5 seconds. Cyclone can be cast on self. Enemy units take 50 damage upon landing.
Cooldown: 23 Seconds
Manacost: 175 Mana
You cannot cyclone allies
Double clicking the item uses Cyclone on yourself
Cyclone is a Normal Dispel, therefore it can dispel certain debuffs such as Dust of Appearance. (More on Dispel)
Meat Hook can stop the cyclone and pull the units without damaging them
Is blocked by Magic Immunity when using on enemies.
Is blocked by Linken's Sphere
Can be Purged
I don't know of any other skills or items that can make you go up in the air yet.