The Ukrainian airforce, which has a limited number of fully operational aircraft to start with has not flown any combat missions for months after the Russians / seperatists shot down numerous planes using increasingly sophisticated SAMS. Likewise, there is always the possibility of direct support from the Russian airforce.
Though there have been some pro Ukrainian partisan attacks on Russian / Seperatist connvoys, the loses trying to capture the airport have been far more than loses from the occasional partisan attack. As the area is largely pro Russian (though not as fervently as Putin claims), the ability of the Ukrainians to mount continuous attacks against seperatist convoys are limited.
Likewise, even if the Ukrainians control and then bulldoze the landing strips, no Ukrainian cargo planes are going to land there. Not only do the Ukrainians have a very limited number of flying aircraft, but landing aircraft would be very vulnerable to seperatist sams fired from Donetsk suburbs. Earlier, the Ukrainians lost a plane and 50 men to sams while trying to reinforce Lugansk airport.
The airport is purely symbolic. As another poster stated, "How much of a "nation" do you have if you dont, at least symbolically, control in the international airport in your capital city?". Also, it is not just the airport,but the surrounding suburbs. The Ukrainians control several Donetsk suburbs. Evidently, Ukrainian forward positions can almost be seen from the roof of the DPR "Parliament" buildings- another potent symbol against the viability of a quasi state.