Here is how to pack a saxophone well:
INSIDE THE CASE
1. Secure all loose items in the case by wrapping them in padding (thin bubble wrap etc) and putting them in a bag in the storage compartment of the case.
2. Put the endplug (a small cylinder about one inch diameter) INTO the top of the saxophone body to protect the octave lever that sticks out to the right side inside the case.
3. Use thin bubble wrap to pad under the horn, especially under the bell, and around the bottom bow. Use enough packing to fill all void space in the case, so that the horn is SNUG in there. If you shake the case, nothing should be able to move inside. Think of the case as what will damage the horn. But also, do not overpack: the lid of the case should still close easily, and the horn should not be so tight that keys could be bent.

OUTSIDE THE CASE
1. Get a HIGH QUALITY box. The best box is the “golf clubs” box at the FedEx/Kinkos stores. It is 48x15x15 double-walled, and can be shortened to fit. The UHaul store sells a number of wardrobe boxes that are similar sizes and quality. Don’t use a cheap or home-made box. It will get crushed, damaging the saxophone. If your horn has no case, it will need to be double-boxed. Use one of the above boxes, and inside pack the horn carefully in a small box that is about the size of the horn.
3. FILL the box TOTALLY full of packing peanuts or other soft foam or sealed-air packing. Do not use wadded paper, or this will invalidate your insurance in most cases. The more peanuts you add, the stronger the box will be. A box that is totally full becomes quite strong and can support a lot of weight. A box that is 90% full crushes immediately.
4. Write the address on the box in sharpie to prevent loss if your label gets ripped off. Tape over your label  so that it won’t get damaged by water.

USE LOTS OF TAPE to tape up the box.

If you do it right, your package will look something like this