Radiation is everywhere, not only around Jupiter. We only start caring about if if the energy levels get too high. There is electro magnetic radiation that is very energy intensive per photon (UV, X-ray and Gamma) the issue here is that, unlike with longer wavelength light, the photons don't just move or exite electrons, but can push them away from the atom or might even be able to punch protons from the nucleus. This kind of radiation ist best delt with with sheelding materials that have huge nuclei and a high density. Think the lead you use when you get an x-ray.
Normal electromagnetic radiation can be dangerous if the intensity gets too high. One can fry an egg on a fighter aircrafts radar. As long as you don't get too close, you'll be fine.
What you care about in the Jovian system is particle radiation. These are particles (electrons, protons, nuclei or heavy ions) that move at a high velocity. This is pretty much equivalent to atomic scale bullets. The way you shield yourself against these is by using materials, which contain a lit of hydrogen. The particles will get slowed down in these materials (water and plastics are great). Be careful however, as this creates bremsstrahlung (literally breaking radiation), so you want a metallic ray shield below the particle shield.
Interestingly a magnetic field can also help against particle radiation, as it deflects the particles away. As long as the particles are charged anyway. Neutron radiation, which is produced by fission and fusion plants, goes through any magnetic field and, depending on its velocity, might make whatever it hits radioactive in turn, pass straight through or punch holes into the atomic structures.
The general takeaway is that while special shielding is great, any large amount of matter you and a radiation source is great. Think the atmosphere of Earth. It shields us against the dangerous UV light the sun emmites.
Also, the link up top is to Atomic Rockets exelent writeup on the matter. Now, onwards to your question.
Jupiter's Wrath
It is a common misconception that the Jovian magnetosphere is representative of gas giant magnetospheres. It isn't. The issue at hand is the innermost moon Io, which spews charged particles from its hundreds of vulcanos. These make the Jupeters orbit especially rough. Not that the other gas giants are harmless radiation wise, but Jupiter is the worst.
If you truely want to settle the system, you're most likely going to clear out the radiation in the long term. This has already been investigated for Earths magnetosphere. It isn't that complicated, it just takes long tethers, which use electric fields to capture the charged particles. Energy is practically free around Jupiter, as you can just tap the magnetosphere with long cables, which move with your stations.
In the short term, you'll most likely use radiation shelters inside water tanks for your ships. If you use certain engines, like minimag-orion drives or anything else that uses massive magnetic fields, these could double as radiation shields. Or vou builed anti-radiation magnetic cages around the spacecrafts. Be careful that the magnetic fields don't become dangerous for people.
As for bases, if you go down to any of the major moons you should just burry or melt yourself down. Under tens of meters of ice or rock (in case of Io) you don't care how much radiation is on the surface. Additionally it is quite likely that we will heve quite decent medication against radiation damage. Damage to DNA is quite important when it comes to longjevity research and this is likely to become a huge area of interest in the next decades.
I'd guess that you'd want resource bases on each of the major moons. The center of the system is in my opinion going to be Callisto, as you can work on its surface and it is likely to have decent mineral wealth.
Orbital bases can also be build into the many irregular moons of Jupiter. These are mostly captured asteroids and comets. You can just hollow them out and build spinning space stations in them. This will be the preferred solution if it turns out that humans don't only have poor health in zero gravity, but also in low gravity (we just don't know yet, zero gee is very from any gravity at all). Additionally you got materials, no gravity which could interfere with construction or manufacturing and a great spaceport.
