How do they ensure there is one electron on an oil drop in Millikan's oil drop experiment? Since air is ionized using X-rays, in the 'observation chamber', there must be several electrons in the chamber, so the oil drops are exposed to several electrons so it seems intuitive that a single oil drop should catch more than one electron.

Then why is it that there's just one electron on a droplet or do they selectively choose those droplets which have a single electron residing on them?
There does not have to be just one electron per drop.  Say you have a drop which, in reality, picked up four electrons, another with five and a third drop with seven.  None has one electronic charge, but when you measure charges you find they have a common factor; the first drop has four times that factor, the second has five times that factor and the third drop shows a multiplier of seven.  It was this common factor, not necessarily the charge on any specific drop, that was recognized as one electronic charge.