Any environment where you breath recycled air increases the risk of infection. You are sharing air with anything up to 300 strangers after all. Tea Tree oil trick (already suggested) is one I have heard but you will need to keep it topped up for the duration of the flight. I am on daily antibiotics anyway (it was for post leukaemia treatment; now combined clarythromycin for COPD as well). This can boost the immune system enough to avoid minor infections.
Be aware also that if it is the asthma that is most affected this may be the dry air and slight pressure reduction of cabin air. Breathing anything with a raised humidity may help anyway. Or discuss with doctor and use a suitable nasal drop?
But I have to confess to not having flown at all now for 10 years - but then I am just a coward...
BTW breathing only through the nose is I have been told, not effective - the nasal set up warms air better but does not have any more notable anti-bacterial properties than breathing by mouth. And you cannot do pursed lip breathing to overcome breathlessness.