I'm going to interpret "surface" to mean Saturn's upper atmosphere.
It was recently discovered that ring particles are raining down on Saturn.
It’s raining on Saturn. Each second, the planet’s rings shed perhaps thousands of pounds of water ice, organic molecules, and other tiny particles into the gas giant’s clouds.
This is caused by particles picking up an electric charge:
Ring particles are caught in a balancing act between the pull of Saturn’s gravity, which wants to draw them back into the planet, and their orbital velocity, which wants to fling them outward into space. Tiny particles can get electrically charged by ultraviolet light from the sun or by plasma clouds emanating from micrometeoroid bombardment of the rings. When this happens, the particles can feel the pull of Saturn’s magnetic field, which curves inward toward the planet at Saturn’s rings. In some parts of the rings, once charged, the balance of forces on these tiny particles changes dramatically, and Saturn’s gravity pulls them in along the magnetic field lines into the upper atmosphere.
This means the rings will eventually disappear, within some 100 to 300 million years.
Other effects on the surface:
- I don't see how the rings could change Saturn's magnetic field significantly. The rings are mostly water ice, which isn't magnetic.