Yes, oxygen (and sunlight) can affect the quality of oil. The oil turns rancid after some time. And storing the oil in a really airtight container (like a can from which air has been evacuated before sealing) should prevent or at least slow the process.
However, the problem is that you can't practically store your oil in an airless container and still use it. Yes, you can seal a bottle of oil. But there will still be air between the oil surface and the stopper, and the oxygen from this air will react with the oil molecules on the surface. And if you use a non-sealed bottle, the exactly same thing will happen. Unlike other cases (soda in a sealed vs. open bottle), the reaction of oil and oxygen isn't quick and aggressive enough to quickly exhaust the small amount of oxygen in a sealed bottle and stop. It will go on regardless of whether there is free air flow from the atmosphere or not.
So what can you do? You can still minimize the rate of rancidification. But it isn't the openness of the bottle that matters, it is the contact area size, because this is where the reaction takes place. Which means, store your oil in a (preferably tall and slim) bottle, as opposed to something like a jug. Protect it from light, because light is definitely damaging for oil. Also, buy it in small sizes, so you can go through it in a reasonably short time. And after all, don't worry too much about it. The reactions are slow, and the oils you use several times per week will not last long enough in your kitchen to deteriorate too much. It can become a problem if you collect special oils which are used seldom and contribute a significant part of the taste in a dish.