They did; The Americans deployed the 82nd Airborne and 101st Airborne in a number of landings, both parachute and glider-based, the night before Operation Overlord began. Additionally, the British 6th Airborne Division (along with 1 Canadian Parachute Battalion) engaged in Operation Tonga.
James Huston's Out of the Blue is a good source for American Airborne operations in WWII, and Lt. Col. TBH Otway's The Second World War 1939–1945 Army – Airborne Forces is great for British and Canadian references.
Airborne forces are very lightly equipped. The allied divisions had some small anti-tank guns and a few artillery pieces, but nothing heavy, and no tanks. They also arrive scattered and have to reconstitute themselves as best they can. The 101st and 82nd had people all over the place, fighting with whichever unit they ran into first. 320th Glider Field Artillery Battalion took off with 12 guns, by the end of the first day only 2 were operational, and the next day only 8. The airborne divisions were intended to be a roadblock to delay German reinforcements trying to reach the beaches. But they weren't intended, nor were they able, to hold off a strong German counterattack from the forces around Calais (who, fortunately, weren't committed to the battle until much later).
Lack of transport aircraft is also a problem. The allies had just enough aircraft to deliver 3 divisions versus landing 9 divisions by sea. And the airborne divisions would have to be resupplied by air, using a dwindling number of transport aircraft due to losses to enemy anti-aircraft fire. Aerial resupply, particularly without airfields, is extremely problematic even if the weather is good. A couple of days of bad weather could leave the airborne troops without ammunition. In order to reliably supply any reasonably sized force, sea transport was required.
So, even if the allies had managed to air drop 6-9 divisions, there was a very strong possibility of a much larger version of Arnhem (A Beach Too Far?). Invasions like Normandy are all about building up combat power faster than the enemy, and aircraft just aren't able to do that on a large scale like sea or rail.