How do Jupiter's moons affect the planet?

Simple answer: not much. Jupiter's moons are far too small to have any meaningful effect on the planet. The largest four moons are about the size of our own moon, and the other moons are pretty much captured asteroids. There is some debate on if the moons have a magnetic field, and if they do, how much that affect's Jupiter's magnetic field. But even that effect would have little impact on the planet itself.

However, the moons do have a great affect on each other. The close proximity of each of the four large moons causes tidal flexing of the moons, in addition to the tidal flexing that they get from being in elliptical orbits with Jupiter. This is the energy that drives Io's volcanoes and also keeps the water under Europa and possibly Callisto's icy surface liquid. In time we may even find that life forms on each moon also depend on this energy to drive their entire ecosystems.

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If E = MC², why do I have less energy the more mass my body acquires?
That is all.

--Azpod... Formerly known as James Justin

In a way, they have had a bigger effect on this planet than on Jupiter. Discovery of Jupiter's moons got Galileo in a lot of trouble and was a death blow for geocentrism (I know! I know!).
Lots of folks working on the problem of finding one's longitude thought that the answer could be found by using the position of Jupiter's moons to calculate the exact time and hence one's longitude. That turned out to be impractical.