I wouldn't call the radiation risks "pitiful." Radiation isn't all the same, and the cosmic rays astronauts would be exposed to are potentially quite dangerous.
Not terribly surprised the idea of launching nuclear reactors from the Earth is a touchy subject, what with the known tendency of chemical rockets to blow up (yes, still happens). You also need to provide for a landing in nearly airless conditions (so parachutes won't be enough by themselves) and enough fuel/propulsion capability to escape the Martian gravity well so you can return home. All so you can get a few meat-bags there for a few weeks to do a tiny fraction of the science that much cheaper unmanned probes can accomplish, given that the probes can stay for years and don't need a return trip.
Again - cosmic rays - thats why you have stormcellars. Big dosage coming along? Everyone into the shielded closet until it's over. Literally one of the first things I said. These are not new or unheard of problems, and mission plans since Von Braun's Mars Project have integrated it into account and evolved with the science.
The probes can't do jack. Most probes die on the ground after a few years, are slow, are incredibly held back in their capacity, and, oh, aren't humans doing anything. Let's say there are fossils on mars, yea? Where are they going to be? Under meters of dirt, even on the seabed, since the atmosphere has been moving around that dirt and covering the fossils for millions, billions of years. No probe can ever dig through that, and we shouldn't keep waiting and hoping that one day we'll make an uberrobot to do what we can do, instead of bringing along bots that, if we placed them on earth in a archeological dig, would serve as coffee tables at best. In the realms of the other sciences, a human on Mars presents massive leaps in transport technology, life habituation, health, and pride; things the probes and landers and rovers barely, if ever, push, much less achieve.
As for the exploding - If a rocket explodes carrying a nuclear reactor or a nuclear thruster, all which happens is that the solid block of radioactive material falls into the ocean, adding barely anymore radiation than the dozen or so rusting nuclear submarines such as the Thresher.
If the rocket blows, to boot. There's quite a few ten-thousand odd launches so far, and the Soviets have already launched a nuclear reactor up there, and it would not surprise me in the slightest if the USAF has or is about to; and the vast majority of launches are successful - especially when we don't rush the engineers to finish their rocket, but give them the time and funding they need and listen to their input - consider the N1 versus the Saturn V, for example. Reasonable chances for me.