It would most likely be safer for you to stay in bed today than go to work or school. But the probability of something happening in transit is small, so you accept that risk. Realistically, that is about the same thing we can say about cell phones.
Radio transmissions operate by electromagnetic fields generated by the transmitting antenna. Even if you don't have a cell phone, there are electromagnetic fields passing through your brain. Those carry the television, radio, and various wireless signals that are part of our modern world. But after a short distance from the transmitting antenna, those signals are relatively weak. Is it possible one of those could cause harm? I won't say it is zero probability, but it is vanishingly small. The probability should be greater for the more powerful signals generated by a cell phone transmitting in close proximity to your head. But, we still don't know the actual probability.
Part of the problem is we don't really understand what electromagnetic radiation does to biological systems. With the exception of microwave heating, the effects are miniscule. It is important to remember that a transmitting antenna does not eject electrons or other particles. Instead it induces an electromagnetic wave which can be detected, but does not consist of particles. The electromagnetic wave could conceivably align the magnetic poles of some DNA atoms, but even if it did, we don't know that it is harmful.
Why you shouldn't worry about cellphones and cancer points out that coffee also increases the risk of cancer. The Forbes article notes that cellphones have been widespread for over a decade, but the number of brain cancers (glioma) indentified by WHO has remained pretty constant. (Glioma is a horrible disease.) So I checked the National Cancer Institute's Surveillance Epidemiology and End Results (SEER) fact sheet. For all causes 22,020 men and women will be diagnosed with cancer of the brain and other nervous system in 2010. The trend has been downward for the 1991-2007 period compared to the 1981-1991 period.
So should you forsake cell phones and walk around with aluminum foil wrapped about your head? Mmm. Lower your risk and use a hands-free (preferably a wired, not Bluetooth) handset.
Also, if you are interested, the CDC also did a study on whether farmers had more brain cancer incidences than the rest of the population. The Upper Midwest Health Study shows they have lower overall cancer risk, but at increased risk for brain cancer.