Saxophone doesn't necessarily take a lot of air, but it does require strong air control. We call it breath control. Breath control allows you to play loudly or softly and stay in tune with a steady sound.
Good breath control, or breath support, starts with the diaphragm. That's the muscle in your abdomen that allows your lungs to fill and empty. When you are able to control that muscle, you can control how fast your air moves through your saxophone. It's a muscle that is exercised with breath control exercises, some of which are practiced away from your instrument. Just like any other muscle, it needs to be exercised regularly to become and stay strong.
One of my favorite breath control exercises is laying on my bed or the floor and placing a book on my belly. Breath in slowly and see how far you can make the book smoothly rise without tipping off of your belly. Then do the same as you breath out. The slower you can raise or lower the book without finding yourself gasping, the more control you will gain of your diaphragm. After doing this exercise take a full breath and then release it as slowly as possible into your saxophone(or any wind instrument for that matter). You will notice an improvement in your tone immediately. This exercise increases your awareness of your breath and diaphragm from a relaxed posture, which means you will be able take more breath into your lungs. More breath = More ability to sustain a long note that sounds good!