A lot depends upon the CME. The CME itself doesn't do much but a fast CME interacts with the slower solar wind to create a strong shock (like a supersonic boom). This shock can accelerate ions in the solar wind up to a significant fraction of the speed of light. These ions, mostly protons, but also heavier ions, in sufficient amounts can pass straight through a spacesuit and into the body of an astronaut. In extreme circumstances the large number of these particles can cause damage at the cellular level and sometimes even result in cancer. It has been estimated that the radiation dose received by an astronaut during a large CME is equivalent to a year's worth of normal dosage. This is one of the reasons why NASA is interested in being able to predict these events. An astronaut on the moon or on their way to Mars would potentially be exposed to a number of CMEs which could cause significant health concerns.

Cheers - David