Yes, other planets have magnetospheres. I'm enjoying studying the Earth's magnetosphere, and, although I have no ambition to follow Neil Armstrong to the Moon, I'd love to travel to study the magnetosphere around other planets like Jupiter, Saturn, Mercury and Mars.
Although it is invisible to the human eye, Jupiter's magnetosphere is huge. It is nearly twenty times larger than the Sun.
One day we might find planets with magnetospheres in other solar systems. That would mean they would be protected from the wind flowing from their own star and life might have had a chance to evolve there.
Here’s a picture of the Moon and Jupiter. You cannot see the magnetosphere though as it’s invisible to the human eye. In the night sky viewed from Earth, Jupiter just looks like a very bright star. But if we could see its magnetosphere, it would look bigger than the Moon does.
Take a look at this photograph of Saturn, showing aurorae at each pole.