Well, in the Roman period at least (and probably prior as well, though I'm not familiar with such artifacts personally) you would use an ear scoop. These were basically tiny spoons that you'd stick in your ear to clean the wax. These could also double as cosmetic spoons for dispensing early forms of makeup.

The Romans had a number of nifty little tools for hygiene. That last picture I linked contains a Roman soldier's kit consisting of an ear scoop, tweezers, and a nail cleaner.

Before the Roman period, people may have used similar tools, perhaps not metal ones, but ones of wood, perhaps bone. Ear scoops continued to be used up through the 19th century, and you can still find similar tools today (though cotton swabs are much more comfortable and less likely to accidentally hurt you).