I doubt if there's any climatological consequences. The average density of the ISM is about 1 H2 molecule/cm**3. Even the densest cloud is only a couple of orders of magnitude above that. That's thinner than the best industrial vacuum we can produce here on earth. It's probably thinner than the atmosphere up at orbital altitudes, and I'm sure the solar wind can even keep that material away from us. The hydrogen formed by the UV breakdown of H2O in our upper atmosphere probably greatly exceeds that from cosmic sources.

It is possible to detect monotomic hydrogen with radio telescopes, but I doubt if it can be mapped to sufficient resolution to extrapolate the sun's path through it.