Holding the alto saxophone is a very important first step that all young students need to carefully work at. Hand positioning greatly affects how you play, so it is important to master it early on. There are three points that you make contact with the alto saxophone that help with the balance:
All three are very important, but pay close attention to your right hand. Think about your right thumb as a brace rather than holding the instrument. Just let the alto saxophone rest on the thumb.
Below is a picture showing you where to put your fingers.
Next, lets go to posture.