There have been several studies looking into health risks associated with air travel, though often, it's the pilots who are the focus of the investigation. A large European study, Mortality from cancer and other causes among male airline cockpit crew in Europe, finds in accordance with others that there is a higher incidence of skin cancer in cockpit crews. This has also been observed for cabin crews. However, they do not find a dose-response, i.e. they cannot demonstrate that flying more increases skin cancer risk, which is what you'd expect if it was radiation exposure during the flight that was causing cancer. Consequently, it is quite possible that it's higher exposure to UV light in their time off work that puts cockpit and cabin crews at a higher risk for skin cancer compared to the rest of the population. In other words this BBC article is overly sensationalist.
Interestingly, mortality from several other causes is lower in cockpit crews, possibly due to selection effects, i.e. you need to be fit and healthy to get the job, and you make sufficent money to afford good health care:
Our study confirms the marked cardiovascular mortality reduction of cockpit crew as compared to the general population. Obviously only very healthy persons are recruited into the profession, and continuous medical supervision may play a role in controlling cardiovascular risk factors. Analyses of mortality by age show that low mortality persists even after retirement. As seen in ESCAPE [=the name of the study] subcohorts and independent studies published previously, aircraft accidents, whether private or occupational cause a substantial proportion of deaths especially among younger cockpit crew. Pilots in early phases of their flight career seem to carry the highest risk of accidental death in aviation. Mortality from motor vehicle and other accidents was low.
In summary, there may be a higher risk of skin (and, apparently, breast) cancer, though staying away from tanning studios and the beach in your spare time possibly alleviates that.
What about non-fatal health risks? Several potential issues come to mind: Exposure to pathogens, stress, and high noise levels.
Exposure to pathogens is of course a concern in an enclosed space, and there are reports of infections spreading on planes summary pdf. When levels of bacteria and fungi have been measured, it was found that they were highest during on/off boarding, when passengers were moving around and when AC was running on anciliary power. In other words, AC on modern aircraft is doing a good job of filtering the air. Coincidentially, there has been a case study reporting rapid spread of the flu when the AC was out of order (see the above-linked pdf). In sum: Wash your hands frequently, don't exchange bodily fluid with passengers, and you should do fine, as long as the ventilation system is running.
Stress is a risk factor associated with many physical and mental health related problems. There are indeed stress symptoms, such as fatigue, headache, concentration difficulties, or gastrointestinal problems reported in this study about a scandinavian airline, though they do not compare stress level/symptoms to the general population. Anyway, not surprisingly, stress levels are lower the higher up on the ladder you are in the hierarchy. Also, the social network helps a lot to mitigate effects of stress. In sum: be the boss, and relax with friends and family, and you'll do fine.
Noise could also be a problem - it is loud on the airplane due to the engine noise. A study investigating the incidence of hearing loss, on probably the same Swedish airline as above, reports no ill effects of noise exposure, however.
Overall, it appears that even chronic air travel leads to little long-term damage. Have a good flight!
If you want to keep up-to-date on research done in the US regarding health risks for flight crews, you can check out this site maintained by the CDC, the US Center for Disease Control.