The head of Ukraine's defence council is one of the founders of the Social Nationalist Party, since renamed Svoboda, the neo-Nazi party... his deputy, in charge of the country's police, is from the neo-Nazi Right Sector. So here we have the neo-Nazis running the police and the prosecutors. I do not blame the Crimeans one bit for taking all measures, including asking for and getting informal Russian military help, to keep them out!
The airports are key places from where Right Sector stormtroopers and troops loyal to the new regime can land to force the will of the new regime upon Crimea, so it is very sensible to have professionals run these places in order to prevent such an occurance. The presence of these miltiiamen also helps stiffen the resolve of Ukrainian police there who, as local Crimeans, would be relunctant to follow orders from Kiev but believing perhaps that they have no choice, doing so and attacking their own people. These miltiias help to encourage them to not follow those neo-Nazi commands.