Have you ever disembarked from a long haul flight feeling "not quite right?" I have. I've never gotten sick from flying, but I almost always get vague discomforts like increased sinus pressure or fatigue. It took me a while to find someone beside myself to blame for this. But, alas, I did.

The Science Blog recently reported on the low quality of air aboard planes:

The cabin air is a mixture of outside and recirculated air. Other substances in the vicinity can add to this mixture, such as ozone, and vapours from hydraulic fuels and engine oils. Combined with a low humidity, these can significantly reduce the overall air quality.

Larger planes are equipped with devices that reduce the levels of ozone, but smaller craft are not. Even with such devices, you are still breathing re-circulated air, which tends to be quite a bit drier than the air you would normally experience on the ground. So some of the discomfort may be due to dehydration.
Remember that guy who flew to Greece despite having drug resistant tuberculosis? I hope, for the sake of humanity, that few people are as selfish or stupid as him. Even so, a case of non-life-threatening common cold can be passed around economy class simply because everyone is in the same place breathing the same air.
What can you do? Probably very little. Taking simple steps like washing your hands often and drinking plenty of water can go a long way. If you want to wear a surgical mask, no one would say no (though you'd probably earn a few giggles). Just make sure you take it off when you try to go through immigration.