Jupiter is a special case. Due to orbital resonances with the galilean moons the inner moon of Io experiences extreme tidal heating, which results in extensive vulcanism (Io's surface is the youngest in the solar system because of this). This creates a sulfur dioxide atmosphere around the moon which gets stripped off by Jupiter's strong magnetic field after which it eventually gets turned into plasma and forms the inner radiation belts around Jupiter.

Without those things happening the radiation environment would be far lower. Saturn, for example, has much milder radiation levels.

Also, life tends to evolve to tolerate whatever level of natural radiation background exists. Most, but not all, life from Earth would have trouble with the radiation levels near Jupiter but life that originated on a moon around a gas giant would probably be fine.

If anyone wants an idea for a scifi story posit a technological species arising on a planet that experiences significant levels of background radiation. Such a species would be able to more easily tolerate working with radioactive materials, as would their ecosystem. For example, something like the Chernobyl accident may not even be a serious problem for them, since every organism would be able to tolerate radiation exposure. They might be able to operate fission reactors without much shielding. And they might be able to tolerate the radiation exposure of interplanetary space without needing any special shielding at all.