You'll always feel the impact of a storm, but that impact scales with the growth of your fleet. There is a talent that will globally reduce the impact 50%, and a hull mod that will further reduce the strike against a ship by 75%.
With a small early game destroyer fleet, you won't really feel the hurt. The typical storm hit may cost around 10-20 supplies or so. Don't fly anywhere with low supplies and you'll be fine. With a late game fleet, the impact is much more severe. I've seen hits around 50-60 supplies, and that's with the 50% talent. Tiny ships will get nuked to 0% with a single strike, giving a form of overall damage cap.
I don't know exactly how the fleet damage scales, but expect to pay somewhere around a week or two worth of fleet upkeep per hit. It will certainly cost fewer supplies to travel a longer way around a storm, but the fuel expense is a different story.