[Editor: A quick google search shows that Mars does have a weak magnetosphere. Check this NASA site on planet's magnetospheres. But you are correct. It is true that because the magnetosphere of mars is so weak, the solar wind has "blown" off most of the planet's atmosphere. In any case, some scientists speculate that in the past Mars may have had a stronger magnetosphere so it's possible that at that time it could have supported life and this life may be hiding beneath the surface or below the polar ice caps.]
Remember that ionizing radiation, such as some of the high energy particles in the solar wind, is a two-edged sword. It not only can kill life, but it can also help create it by increasing mutation rates and promoting the formation of unstable molecules. The Urey/Miller experiment simulated parts of the rather raw conditions on a new planet to show that organic molecules not only could form, but that exotic conditions (ultraviolet radiation, electrical discharges, etc) *helped* them form.
Of course, life needs a place to hunker down where it is protected from the agents that may have helped its birth. Water and earth are great shields, as is an atmosphere -- at the surface of the Earth, we have the equivalent shielding of 33 feet of water above us -- all of this just from the atmosphere. You can stand at the top of an operating nuclear reactor, looking down through 20 feet of crystal clear water to the glowing blue Cerenkov radiation produced from high energy particles being rendered harmless by the water, with no real danger to yourself at all.
But on planets with no significant atmosphere, a magnetosphere is a great help. On the Moon, which also has no magnetosphere and is outside Earth's, future astronauts will have to dig into the soil to build shelters for themselves if they plan to stay for sizeable durations and endure radiation storms produced by solar flares.
Some additional commentary: A magnetosphere provides protection to the surface of a planet, but generally, these particles play only a small role in stripping the atmosphere from a body. Venus, for instance, has no significant magnetosphere but has retained a much thicker atmosphere than Earth. Gravity is the primary thing that holds an atmosphere down. Still, lighter particles can leak more easily into space. Even Earth's gravity can't hold hydrogen very well. Fortunately, hydrogen in Earth's atmosphere is mostly bonded with oxygen to form water vapor, and the larger mass of the water molecule keeps most of the planet's hydrogen from otherwise leaking into space. On Venus, one proposed explanation for part of the puzzle of its different evolution is that life did not evolve early, or conditions were inhospitable to its formation, so oxygen did not get generated into its atmosphere through biological processes. Without molecular oxygen, there was no generation of ozone high in Venus's atmosphere by interaction with ultraviolet light. Instead, it is thought that this energetic ultraviolet light passed into lower and wetter parts of Venus's atmosphere where it could break apart water molecules, liberating hydrogen. Over time, this hydrogen would leak out into space, leaving Venus's atmosphere ever drier and more inhospitable to the formation of life. On Venus, the race was lost. Was it a close race on Earth?