Everybody seems to have some advice, not all of it is qualified or based in fact. There is lots of good information floating around the Internet, such as The Saxophone Buyer's Guide by Jason DuMars, so I am not going to rehash the advice of others. Just a few more things to consider from my experience and possibly a new perspective:

Set a budget. Try to buy at the upper end quality wise of your budget range rather than the lower end of the next level of saxophones. You may be able to purchase a tired or beat pro horn for around the same price as a new student model, but repairs to saxes are not cheep so beware when buying a second hand sax. ask if you can take it in to be looked at by a repairer and get an evaluation or quote before you decide.

Choose a name brand. For beginner student models,  I tend to recommend second-hand Yamaha horns as being the most reliable lowest entry level cost saxophone. Very few beginning players of any age stay with their first horn forever. Buying a name brand gives you a far better chance of a reasonable re-sale price when the time comes that you want to upgrade.

Be guided by physical appearance. Lots of dents, worn lacquer, a ratty case, all indicate neglect or even abuse that can result in repair bills which turn your bargain into a disaster.

Preferably get a teacher or experienced player to test play the horn you are considering. If this is not possible, visually check the pads. Dark dry pads are probably not sealing well and will require replacement very soon.

Check the condition of the mouthpiece. Look for chips and cracks or "home modifications". Once again get an experienced player to assess the suitability of the mouthpiece for the player. A good standard beginners mouthpiece can be purchased from around $50.