yes, they do!
For instance, solar storms can cause aurora (or northern and southern lights) here at the Earth. These are also seen on other planets, like Saturn,
http://www1.nasa.gov/home/hqnews/2005/feb/HQ_05049_saturn_aurora.html(although I should say not all aurora are related to the solar wind - Jupiter's auroras are caused by interactions with Jupiter's moon Io).
The Earth and Saturn both have relatively large, strong magnetic fields, but some other planets, like Mars do not, so the effects of a solar storm might be different there. Here on the surface of the Earth the Earth's magnetic field protects us from dangerous high energy particles that come with solar storms. The surface of Mars would not have that protection.
Terry