The answer is pretty much no. Life on earth exists without a apparent problem in places where the background radiation is hundreds to a thousand times higher than average. Since cosmic interstellar radiation is only a small portion of the total radiation life recieves on earth, radiation from space would have to be whoppingly huge compared to now to have an effect. Then there's the fact that life might be quite happy develping under a rock, or under an ocean or beneath an icecap where it would be well protected. We also know life can develop resistance to radiation.