So, there are two aspects to this problem. The first is one that we deal with today on a regular basis, that of the Doppler shift of a spacecraft orbiting the Earth, or otherwise moving towards/away from the ground-station. This is the same phenomena as the Red Shift, just at a lower magnitude. Why is this important? For a 150 MHz uplink, the Doppler shift is around 3 kHz for a satellite in LEO. That actually is a channel in some modes of communication. The higher the frequency, the larger the jump, such that an FM channel could entirely be passed, if the doppler is not taken in to account.
So, how do you correct for that? Basically, calculate what the doppler should be, and send at a frequency such that the spacecraft will get the right signal. Additionally, you can build a control circuit such that the detected frequency window can drift slightly, according to the carrier frequency.
As for the second problem, that of radiation becoming higher energy as you approach it. How is this managed? First of all, let's see where it would become an issue. Let's assume a frequency of 10^15, which is in the realm of visible light. The energy becomes ultraviolet at an increase of about 10 fold frequency increase. To do that, you have to be moving at 0.9c, thus, you are hit by the wavelength 10 times more (1/(1-speed)), giving you an increase of 10 in frequency. That is a very significant speed. Furthermore, this will only be coming from the front half of the spacecraft, the back half will see a 50% decrease in frequency, which is harmless (See this calculator)
So, what else is a spacecraft going to need to do that is moving that fast? There must be a way to protect it against small particles, even the most minute particle would cause issues at that speed. That would involve some physical shielding, no doubt, on the forward site. In addition, there is high amounts of radiation anyways in space, which will need protection.
The bottom line is, radiation from moving this fast isn't likely to be a problem. It will likely be solved when dealing with other issues the spacecraft will have.