A stable atom in the immediate vicinity of a fission reactor can become radioactive if its nucleus absorbs an energetic particle. The dominant energetic particle produced by the fission reaction is the neutron.
When a neutron hits a nucleus it may simply bounce off it, imparting some kinetic energy to the atom. This can cause structural defects in solids, and is an important issue in reactor design, since too many such defects can seriously weaken the materials that the reactor is constructed from. But of course, this isn't an issue for gases or liquids, the energy will merely raise their temperature.
If the neutron doesn't bounce but is instead absorbed by the nucleus the atom is transmuted to a heavier isotope of the same element. In some cases, the new isotope is also stable, but in other cases it will be unstable, that is, radioactive. For atoms with a small atomic number (Z, the number of protons in the nucleus), the most stable combinations have an equal (or almost equal) number of protons and neutrons. If the proton : neutron ratio deviates from this then reactions occur to try and achieve a more stable ratio.
If a normal $^4He$ nucleus manages to absorb a neutron it transmutes into $^5He$. The probability of this happening is very low, but even if it does happen there's nothing to worry about because $^5He$ is very unstable: it decays back into $^4He$, with a half-life of around $7\times 10^{-22}$ seconds, emitting (of course) a neutron. So the end result is virtually indistinguishable from the neutron simply scattering off the nucleus.
Free neutrons themselves have a half-life of around 10.3 minutes, decaying into a proton, an electron, and an electron antineutrino, so it's possible that our helium nucleus gets hit by a proton. But it's much harder for a proton to be absorbed by a nucleus because the positive charges repel each other. It normally takes huge energy (high temperature) for such nuclear fusion reactions to occur. And even if by some miracle a proton is absorbed by a $^4He$, the result is the highly unstable $^5Li$, which has an even shorter half-life than $^5He$, and which decays by emitting (you guessed it) a proton and reverting back to $^4He$.
There is actually another stable helium isotope, $^3He$, but it's very rare, and of course if it absorbs a neutron you just get regular $^4He$.
So in summary, when helium is in a fission reactor it cannot become radioactive, it will just get warmed up, which makes it very useful as a coolant.