For typical X-rays of 20 keV, I found a density attenuation coefficient of 0.0757 m²/kg in air. Considering a typical air density of 1.2 kg/m³, this gives an attenuation coefficient of 0.09 m⁻¹. The inverse of that - around 11 meters - gives you the distance, after which an X-ray beam of 20 keV is attenuated to e⁻¹ of its original intensity. Taking 100 times that distance - 1.1 kilometers - results in an attenuation to e⁻¹⁰⁰. That's a number, that starts with zero and has over 40 more zeros after the decimal dot, before another digit comes. So it is safe to say, that the atmosphere is fully intranspartent to 20 keV photons.
At 100 keV, the density attenuation coefficient of air reduces to 0.0155 m²/kg. Therefore, the average length for an 1/e attenuation increases to around 50 meters. That's still not enough to "see" those hard X-rays / soft gamma rays through the atmosphere.
At 10 MeV, the coefficient further reduces to 0.00205, which gives an 1/e distance of around 400 meters. So, again, virtually all of those gamma rays are absorbed by the dense atmosphere between ground level and 3000 meters height. 11 km above ground, the typical flight head of aircrafts, the air density is reduced to 0.36391 kg/m³, resulting in an 1/e attenuation for a 10 MeV beam of 1340 meters. That gives you a tiny change to be hit by 10 MeV gamma quants, that are traveling towards earth from outer space, or which are generated from particles of the solar wind, that hit outer atmosphere at 25 km altitude and up.
On the legendary Concorde - 18 km flight height and surrounding air density of just 0.1 kg/m³ - the 1/e length further increases to almost 5 kilometers. As air also quickly thins out at higher altitude, those 10 MeV quants coming from above and directly heading down have a high probability to reach the Concorde and its passengers. The good news: Being so energetic, most of those 10 MeV quants will just hit through and not deposit much energy on the aircraft and its passengers.
Particles of TeV energy are strong enough to penetrate atmosphere and hit the ground. But as those particles are rare, direct hits to counters on the surface are very rare, too. But as they penetrate through atmosphere, they constantly deposit energy, namely through Czerenkov radiation, and that energy can be measured at nighttime. This makes the earth's atmosphere basically one huge Czerenkov counter for cosmic TeV particles.
Good reading starts are here:
So, yes, SOME cosmic radiation can be detected through the atmosphere at ground level. But for X-rays and low- to medium-energy Gamma rays, the atmosphere is intransparent, and one has to go to space (or to very high flying balloons) to measure them.