I'm aware your question regards Saturn but one thing I use to keep hearing is that 'Jupiter was nearly a star' which was always as quickly dismissed (this idea was even played on in 2010). If I'm not mistaken, Jupiter would have had to have been at least 10 times its current mass to have become a brown dwarf (though its radius would have remained approximately the same). The difference between brown dwarfs and large gas planets is that gas planets have a metallic core and the outer layers of gas remain seperate; with a brown dwarf, the metallic core mixes with the various gas layers through convection, material rising to the surface, cooling, and returning to the centre (on a side note, fusion in lower mass brown dwarfs doesn't necessarily take place, brown dwarfs bigger than 13 Jupiter masses fuse deuterium and those bigger than 65 Jm fuse deuterium and lithium). In order for either Saturn or Jupiter to have become brown dwarf stars, there might have been a need for at least 10-15 times as much material orbiting the sun as the solar system formed, I imagine this would have had some effect on the formation of the other planets.
Considering the idea of Jupiter being a brown dwarf star is a nice idea though as the various moons such as Europa and Ganymede would warm up, the ice would melt and atmospheres might form. The fact that the temperature of a small brown dwarf is in the range of about 900 to 1500k means the moons shouldn't be obliterated as the would by a regular star capable of fusion. It would be interesting to think of what would become of Io. In this case, not only would our solar system have 2 suns but we would have something that resembled a mini solar system within our own.
regards
Steve
