Mineral oil is also non-polar so it will prevent water from attacking the surface of whatever is stored under the mineral oil. (Since the water will not mix with the oil). Oxygen is also less soluble in mineral oil because of the non-polarity. Oxygen has a couple of unpaired electrons on it (when in an elemental form) and those electrons have a negative charge. While the molecule itself is not really polar in nature, those unpaired electrons will want to seek out a positively charged substance. Water is a very polar molecule, so those electrons have a bit more of an attraction to the water molecules than it does the non-polar mineral oil. As a result, oxygen dissolves in water to a greater extent than it does in mineral oil. So by putting an oxygen sensitive compound in mineral oil, you lower the amount of oxygen it can react with by covering the surface with the oil, and not having as much dissolved oxygen in the oil. Now this doesn't mean that oxygen doesn't dissolve in mineral oil. It certainly does and this can be seen with alkali metals. My potassium was freshly cut on one side before being placed in the mineral oil it resides in. Over a period of about 7 months, the oxygen that slowly dissolves in the oil has oxidized the surface to a noticeable extent.