There are three "balance points" that hold the saxophone when it is being played: The LH thumb on the thumb pad or octave key, the RH thumb in the thumb hook, and the neck strap. Let the neckstrap support the entire weight of the saxophone and balance and control it using the thumbs. Of course the remaining point of contact that also helps to secure the instrument is the top teeth pressing down on the top of the mouthpiece. As on the flute, the fingers "play" the instrument, not hold it. If you find yourself needing to put down fingers to balance or support the saxophone, there is something about your holding position that needs to be corrected.
An additional thought on the title of this thread. Fingers that are not being used are ordinarily slightly curved and held close to the keys. Exceptions are when the hand leaves it's normal position such as when playing the RH side keys, and playing the LH palm keys depending upon the size of your hand and length of your fingers.