Why do some gases transfer radioactivity and some don't?
Chemically, helium is inert because it has a "filled valence shell" of electrons, which is very stable; it's extremely difficult to change this structure, as doing so requires a lot of energy and produces a system which is likely to quickly revert back to its ground state under normal conditions.
The helium-4 nucleus is in a very similar situation: in a sense, it has "filled shells" of protons and neutrons. Relative to its neighbors on the nuclear chart, it's one of the most stable nuclear configurations we have measured. Changing this nuclear structure in any way is difficult, so helium-4 is unlikely to become radioactive in the first place, and the configurations that are created when it does happen are so unstable that they decay almost instantly. Neutron capture (far and away the primary cause of secondary radioactivity) creates helium-5, which decays with a half-life of $7\times 10^{-22}$ seconds, so it barely even exists, and certainly won't be found outside the reactor. It's also basically impossible to excite the helium-4 nucleus to a higher energy level using gamma radiation from a fission reactor, as the next energy level is 20 MeV above the ground state (for reference, most of the steps of the uranium decay chain have a total released energy of only 4-7 MeV). So it's safe to say that helium-4 is radiologically inert.
The term "conduction" is probably* referring to the following process: a radioactive nucleus predisposed to emit neutrons decays, and the emitted neutrons are captured by another nucleus, which might make it unstable and therefore radioactive. In this sense, what determines how readily a substance "conducts" radioactivity is its willingness to capture neutrons (aka the neutron capture cross section), which is heavily dependent on the specific nuclear structure. (There are other ways to induce radioactivity, like beta decay of one nucleus followed by electron capture by another, or gamma-ray emission and absorption, but the conditions required for those processes are rarer.)
For other radiologically-inert substances, one might look for other nuclei that have "filled shells" of protons and neutrons. In nuclear structure, these are called "doubly magic" nuclei (having a "magic" number of protons and a "magic" number of neutrons), and do indeed have a reputation for stability, though none are quite so stable as helium-4. Doubly-magic nuclei include oxygen-16, calcium-40, and iron-56.
*Let me stress that "conduction" is highly nonstandard terminology; the term for the process that I describe here is "induced radioactivity."
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.
elements do not "conduct" radioactivity. some are capable of being transmuted into radioactive isotopes of the same element or new elements that are radioactive, upon being exposed to intense radiation by neutrons. Helium happens to be resistant to transmutation and hence does not become radioactive upon being irradiated.