That is true.
But wouldn't the higher energy "blue" cosmic rays be offset by their lower energy "red" counterparts (the ones you are running away from) with it all averaging out to same thing you would see at zero velocity. This would depend on the source of course. I'm assuming uniformity in three dimensions. You may be talking about cosmic rays from the sun or some known source. Well that would be different. In that case I would agree with you.
It still seems to me that if the source of the cosmic rays is not known and the incoming rays are independent of angle then you should not be able to make any measurement on these rays that would tell you anything about your velocity. If the energy of the rays in the forward direction is higher than the rays coming in from your rear then a measurement of this energy would tell you what your absolute velocity is. Except that there is no absolute velocity. Of course if you knew the source of the cosmic rays then you could obviously make a measurement of your relative velocity with respect to that source.
This relativity stuff is really difficult.
I mistated the nature of cosmic rays. They are not massless photons, they are particles.
So at that, a person in space is more exposed than a person under the blankets of magnetism and atmosphere that cover earth. I suspect velocity relative to earth is an inconsequential variable in calculating exposure to cosmic radiation.
thus, in the first case, they are being emitted at a lower frequency towards us, and we are hitting them at speed, raising the apparent frequency from its lower frequency. In the second case, they are being emitted at higher frequency, and we are pulling away, which lowers the freqency...thus, in a universe where both the emitter and receiver are moving, the results are rather ambiguous.