Hi,
That is an excellent question. Actually it is mostly the magnetic field of the Earth that protects us from solar storms. If the Earth did not have that there would be a lot more problems with radiation exposure to airline passengers, power-line surges, and computer glitches on our satellites. We'd probably spend the money to make spacecraft components (computers) that are more shielded or less sensitive to the radiation; put more shielding on airplaces; and the power companies would probably store their energy in different ways or over more locations.
If the Earth didn't have an atmosphere (but did have the magnetic field) the solar storms wouldn't be our biggest concern. Other than not having air to breathe (a big problem), we'd also be very exposed to the UV light from the Sun (most of which is currently absorbed by the atmosphere) which would cause burning (sunburn) and extreme hot and cold temperatures (depending on whether you were in the shade or not). We also wouldn't have rainbows, aurora, or colorful sunsets/sunrises.
Best,
Christina