>Just running into cosmic dust at the speed of light would not be fun.

That's why you need a navigational deflector dish. http://vignette1.wikia.nocookie.net/memoryalpha/images/b/b4/...

Isn't that the whole relativity theory: that if you emit a photon while at celerity 1c, it looks like it goes away at 1c both from you and from a still reference? hence blue is still blue at 1c?

The spectrum of light depends on the frequency of the vibrations in the electromagnetic field, not the velocity of the wave. For both red and blue light the wave is traveling at 1c but the peaks of the wave are at 700 nanometers or 400 nanometers depending on the color. And under Special Relativity distances distort so that you do in fact see light shifted to higher frequencies/shorter wavelengths in the direction you're traveling.

I'm talking about radiation from other stars or what ever. Imagine if every star in the star field in front of you shifted from visible light to gamma.

I'm sure there is a study out there that describes the exact amount of radiation exposure you would receive at reletivistic velocities due to blue shifting.

The speed doesn't change but the wavelength does. That's why we see light from other galaxies redshifted.