Start with the types of airports. Civil vs. military. Then civil taking into account population, flights per year (as a function of wealth of the population), capacity of given airports (large, medium, small). Large airports = 4 runways, medium airports = 2 runways, small = 1.
Military would be something like the number of countries with airforces, how many planes per.
This is a good top down one :) First clarify if they mean major airports or even smaller regional ones, but I assume they mean all airports
Think about breaking this down by continent/country. By each major region, estimate # cities with major airports (i.e. US ~80, Western Europe ~50). Once done, think of a pretty large multiplier (20x) for medium sized airports, and then an even larger multiplier for local airports (100x).
You can use a top down approach as suggested by other experts, but then you could do a quick sanity check of your result using a bottom up approach.
You can start from a region that you know and calculate the ratio Airport/population. Then you can scale this number to the world population making some assumptions (e.g. I wouldn't use the same ratio for African countries).