Quantitatively, how much would radiation levels increase without the geomagnetic field? Many, many popular science articles claim that if the Earth didn't have a magnetic field, then the much higher concentration of cosmic rays that reached the surface would cause health damage to humans. (It's easy to find a zillion such articles with a simple Google search, so I won't bother linking to them.) Some of these article just say that the higher concentration of radiation would be "dangerous", while others use stronger language like "catastrophic". But I couldn't find any articles that give any hard numbers, or even qualitative language more specific than broad terms.
If the Earth's magnetic field vanished overnight, then how much more radiation (in grays or sievert or whatever quantity this can be most easily answered) would Earth receive at sea level?
My guess is that the health issues wouldn't be that serious, since the Earth's magnetic field has changed orientation many times and left the Earth with a negligible magnetic field for thousands of years - and as far as I know, those events didn't cause mass extinctions or anything. (Although I believe that the last flip occurred before the evolution of anatomically modern humans, so we don't have any direct  evidence about human life in the long-term absence of a geomagnetic field.)
A related question is about the claim that without a geomagnetic field, the stronger solar wind "would" or "could" or "might" (depending on the pop-sci article) strip away the Earth's atmosphere. If this is true, then I'd be curious what the time scale would be. It must be geologically long, given the many periods of thousands of years without a geomagnetic field during which the atmosphere was apparently negligibly affected.
 A: There is no expectation of any catastrophic biospheric effects from the loss of the Earth's magnetic field. The increase in human ionizing radiation dose would not be significant.
The current world average radiation dose from all natural sources (radon/thoron, internal, terrestrial, cosmic) is about 2.4 mSv/year. According to one study, even in the worst case, losing the Earth's magnetic protection would only increase the typical cosmic ray radiation Earth surface dose from 0.3 to 1.2 mSv/year, i.e. the total average dose could increase from 2.4 to 3.3 mSv/year. This is far smaller than natural geographic variations in background radiation.  For example, the average natural background radiation dose in Florida is only 1.3 mSv/year, while in South Dakota it is 9.6 mSv/year.
If you live in northern latitudes, the biggest radiation risk from losing the magnetic field might actually be from increases in UV-B radiation due to enlarged ozone holes caused by increased high-energy proton fluxes in the upper atmosphere.
It is not clear if the net effect of the Earth's is to protect the Earth's atmosphere from erosion by the solar wind.  Losing the Earth's magnetic field would increase some atmospheric losses but decrease others.
The magnetic field protects the atmosphere from direct interactions with the solar wind, but a large magnetosphere also collects energy from the solar wind that can heat up atmospheric ions.
Venus and Mars do not have geodynamos, and their atmospheric masses are respectively two orders-of-magnitude larger and smaller than Earth, but all three planets have very similar atmospheric escape rates. Losing the Earth's geodynamo should not have any large effect on the Earth's atmospheric erosion rate.
Note that unless you stop the solar wind, it is not possible for all protective magnetic fields around the Earth to disappear. Turning off the Earth's geodynamo is not sufficient. This paper on "Solar wind induced magnetic field around the unmagnetized Earth" suggests if the geodynamo stopped, the solar wind would quickly induce a magnetic field that could provide similar protection from cosmic rays. This is similar to the induced magnetosphere of Venus.
The only obvious way to stop the solar wind is turn off the Sun, in which case worries about changes in cosmic ray radiation doses would not be our biggest concern.
A: The geomagnetic field is most effective at shielding the equator, and not very effective at shielding the poles. So we see aurorae frequently at the poles, rarely at the equator. The aurorae represent radiation blocked by the atmosphere, not the geomagnetic field. They don't extend to the surface, so the atmosphere is obviously blocking most the the radiation. There is not much latitude variation in cosmic radiation at the surface. Thus, we may conclude that the atmosphere is highly effective at blocking cosmic radiation, and we should not expect much variation with changes in the geomagnetic field.
