Elon Musk just announced his concept for traveling to Mars. During the Q&A he commented that the risk of radiation danger to the passengers was low. I seem to remember concerns for the astronauts during the Apollo era if flares occurred during the passage to and from the moon. Would there not be risks of multiple incidents during the longer flight to Mars? What levels of exposure might be considered reasonable especially considering these Martians will also have greater lifetime exposures because of the limited protection Mars affords them throughout their lifetime on the planet? How great is the risk traveling in space for months during the trip with the risk of exposure from cosmic sources and multiple solar flare events?
If you just stay here on Earth, you have an 18% chance of dying of cancer. Let's first consider a one-way trip, ignoring solar flares and radiation after you land. So just consider the cosmic rays, whose flux is pretty constant and which cannot be reasonably shielded against. If you go to Mars in six months, you'll get 0.3 sieverts, increasing your chance of getting cancer to about 20% or 21%. That's actually below the OSHA career limit of a 3% increase. So that would be considered "safe".
On the surface of Mars, the atmosphere provides a very good start at shielding, and you can pack a bunch of dirt on your habitat to significantly improve on that. If you're on Mars for only a year and half waiting for the return trip, you can keep the excess cancer pretty small. Though if you're going to Mars to stay for the rest of your life, it would be difficult to put up enough shielding to avoid shortening somewhat the rest of your life. To replicate the shielding we have from our atmosphere here, you would need to be under something like 20 feet of dirt.
If you return, giving you another six months in space, you have now exceeded the OSHA career limit, increasing your chance of getting cancer 20 years later by 5% to about 23%. Oh no! Clearly going from an 18% chance to a 23% chance of getting cancer is much too high a risk to accept in order to be the among the first humans to walk on Mars!
There are other effects, but excess cancer fatalities are the largest given our current understanding. Here is a good article on the subject.
As for big solar flares, those things can definitely kill you if you're not prepared. Forget cancer. This could result in near-immediate death from nervous system damage. You are pretty unlikely to be hit by one. However, a spacecraft that I am very familiar with that was on its way to Mars in 2003 got hit by a giant solar flare that would definitely kill a human with only typical spacecraft walls to shield them.
The good news is that a) you can get hours of warning if you put some spacecraft around the Sun to watch for the events, since the protons move much slower than light and x-rays, and b) you can reasonably shield humans in a spacecraft from the protons. Generally water, food, and human waste will make for good shielding. You can have a small storm shelter that the crew can crowd into for the duration of the event, which could be on the order of a day or two. The shelter needs shielding in all directions. Since the protons in the plasma are moving much faster than the cloud is as a whole, they are coming at you from all directions.
So, yes, there is some small increase in risk from the radiation. Small, assuming that you are prepared for flares. However that risk pales in comparison to the many other risks of journey, like surviving launch, surviving landing on Mars, or surviving every day, day after day, either in space or on Mars, always with a fraction of an inch of material between you and instant death. Or the risk of losing your source of heat and freezing to death in your otherwise safe habitat on Mars, where it is way, way colder than Antarctica. Or losing your means of producing oxygen. Or drinkable water. Or food. I could go on.
I wouldn't sweat the radiation risk at all until those other risks get down to the percent level, which could be quite a long time.
There was a sensor called RAD (Radiation Assessment Detector) on the Curiosity rover, for measuring the inflight radiation does.
On the one hand, the results showed that there was a significant radiation dose in flight. On the other hand as Robert Zubrin notes, there was a similar sensor on an earlier flight that showed the same basic levels, and then it was evidence it was safe, and now it is evidence it is not safe.
Zubrin also notes that the radiation dose is almost identical to that the ISS astronauts take if they were to stay in orbit for similar amounts of time.
Thus there is some debate on this topic.
Solar Flares which are heavy protons and slower than light, (so when detected at Earth or a satellite closer to the sun, there is time to react even when messaging at light speed) can be stopped with several inches of water. In that case, a storm shelter aboard, where the water/food are layered around it, can suffice.
High energy photons/gamma rays/cosmic rays are harder to shield from, and just have to be accepted as risk.