No one planet is always closest to Jupiter, as the range between Jupiter and the other planets varies depending on its position in orbit. The closest approach Jupiter is capable of making to another planet is with Mars, at a minimum separation of 551 million kilometers.
When Mars and Jupiter are on opposite sides of the Sun, their maximum separation exceeds 1 billion kilometers. During these periods, Mercury, Venus and Earth are all closer to Jupiter than is Mars. Sometimes, when the closest approach between Jupiter and Saturn coincides with the maximum separation of Jupiter and Mars, Saturn becomes the planet closest to Jupiter.