Either you are not supplying enough power to your ships or you are taking too much radiation dmg. Raditation damage itself is reduced my researching the shielding/reduced dmg trait. I believe it's the leftmost tab in the research buttons. Same place you research reduced hyperspace costs. Keep in mind that will only appear once you build an anti-radiation field on a ship somewhere. As for the power problem which can also cause ship damage I believe, you just build a power station and toroidal magnets as are needed to supply your fleet with power if your hiigaran. You can add me on steam if you have other questions etc.

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Also you should be able to see where the damage is centralized in the zoom out map. Will be indicated by a slightly reddish purplish line where the radiation is strongest. You can either reduce or nullify the damage you take by avoiding those areas. But as I said before you can only fully take care of it if you do the appropriate research and build anti-radiation fields. Remember that every map has an intrsinsic radiation hull dmg value. So not every map will require the same amount of research or shielding to save your ships.

edit II

Enemy ships do get affected. They probably had the requisite research or were travelling with a ship that can protect you from radiation from the start. The motherships are known to be able to protect other ships from radiation if they are within range of it. I believe it starts the game with some kind of field so you don't start taking hull damage the moment you load in. It's quite big too.