That civilization would rely more on local power generation.
As Adrian Colomitchi mentioned, shielded DC current would work but DC has higher losses over distance. This can be mitigated by thicker copper wires and increasing the voltage.
As a result, each town would likely have its own generator(s) and long distance transmission of power would be very expensive. Remember that there were community steam plants that would pipe steam to neighboring buildings. DC electricity through decently sized wiring would have less of a power drop over distance than steam and would be seen as a good replacement.
They may even develop the battery technology to store the power generated in long wires. Eventually the storms may be seen as bonus energy.
The telegraph which wouldn't happen for obvious reasons. Though it might be used when secure communication is absolutely needed and the cost of shielding a cable that long is seen as worth the cost.
Radio would still work, it would just have interruptions and people would not rely on it for continuous communication. Morse code or something similar would likely develop. Messages would likely be queued up to be sent when the "weather" was clear. You could have relay stations that would be manned by people who would receive messages on one frequency and then pick the frequency of the next stop for that message and send it to the tapper (the one who sends the messages) for that delivery location. That would involve a lot of people at each station to receive, deliver and sort messages. That sort of thing happened on a smaller scale with the telegraph.
The first computers would likely be electro-mechanical linkages to receive messages on punch tape and those tapes be queued up for the mechanical tappers. The receiver would punch on the tape for the next hop in the delivery and that tape would spool until it was able to be delivered. The receiver would probably use some kind of parity code to make sure it got a whole transmission. The sender would then re-transmit until it got a good parity code back. This would eliminate the need to know when the storms would be over and would allow some messages to go through during lulls in the storm.
This bit of mechanical computing would cut a room full of 10 to 30 people to 1 to 3 people as you would just need to make sure that everything is oiled and to fix tape breaks.
The only real difference would be the general slow down in tech development. Communication may be voice or handled as above. The astronauts would be much more on their own and rely on more simple signals rather than voice transmission.
The need for shielding may be seen as obvious from the start, though the first fried astronaut would tell them that. Manned missions would be much more expensive and not likely to happen without very good "weather" prediction. Unmanned missions would still be very possible.
There will actually be a big push for the space industry: communication satellites. It is much faster to bounce a signal off of a satellite once during clear "weather" to go from coast to coast than to bounce it dozens of times looking for clear "weather".