Imagine that some kind of radiation coming from space would hit the earth tomorrow and would stop every communication on our planet. The internet would stop working, as well as every other kind of communication like wired and wireless phones, every kind radio signal etc. . Every bank accounts would stop existing, military communication would shut down etc. What would happen to society? Which other consequences would arise?
The premise of your question is flawed
It would mean that you have such a huge radiation density that it would most likely severely effect any living thing.
Blocking (or jamming) wireless transmissions is not that difficult, but already takes a lot of energy if you do it locally. If you want to block every wireless transmission everywhere that'd be insane amounts of energy being pumped onto the earth. Probably more than the suns, since the atmosphere blocks most of that radiation.
If you want to block wired transmissions you'd need even significantly more than that.
You would need insane amounts of radiation on every frequency from 1Hz to 100GHz. Most of the frequencies are not very reactive with biochemistry, but some are. And also the frequencies used in current day technology are effecting biochemistry. Most of those are used so little no actual harm comes from it, but the amount of energy your proposed radiation would mean would destroy a lot. And it would also boil water, probably. (WLAN: 2.4GHz, microwave: 2.455GHz)
I think even if you would ONLY have EXACTLY the frequencies we use for communication the side-effects of the radiation intensity on biochemistry would still be destructive.
Society would probably not exist anymore since everyone dies of cancer, radiation poisoning and other things.
If we ignore that then there is no further answer to be provided, since the leftover question of what would happen to society is very opinion-based.
A more realistic scenario would be a world-wide EMP. But the problem of the question being mostly opinion-based is still there.