Keep in mind you may or may not have contracted the illness on the flights, even if you started feeling sick during or shortly afterward. Depending on what has caused the respiratory infection, the average incubation period can be as long as several days following exposure.
Generally I find practicing good hygiene, staying hydrated, and getting adequate rest (as much as possible) throughout the trip is best, rather than focusing just on the flights. We do use appropriate wipes to clean off certain surfaces on the plane, hotel, etc. though that's not a panacea. As someone with chronic sinus issues, non-medicated saline rinses as mentioned above are an excellent thing to use regularly, not just to guard against infection but also to keep dust and allergens flushed out, too. Can't hurt to get flu and other vaccines as appropriate, too.
Finally there's always an element of luck involved no matter what...or bad luck, as the case may be.