My saxophone has been leaking spit all over the first three keys that you put your fingers on (don't know the names of them) and I'm worried that it will damage the pads. My parents are renting it used and it has a couple problems like the octave key not working on high A and above every once and awhile, 1 or 2 screw coming loose, and the spit problem. The main thing I'm worried about is if those pads are getting damaged by the almost at this point constant spit all over they pearls and pads.
As said in the comments, what you observe is likely to be condensation, which is a normal phenomenon on a sax.
However, it is a good habit to clean it with a pad saver after you played in order to remove the moisture from inside the sax, preventing corrosion and eventual pad damaging… After cleaning with the pad saver, do not let the saver inside the sax as it will retain humidity inside the sax and prevent it to dry properly…
If you have the appropriate screw driver, fastening a bit these screws may prevent you from a bad surprise, like a bar falling off or something like that ;).
You should not be able to see fluid, be it spit or condensation, coming out of the upper holes. I wonder whether the pads are closing correctly.
If in fact you feel air escaping from your lips (not going down the saxophone "tube"), then you have embouchure problems - might benefit from lessons and/or a softer reed.
Similarly, if screws are working loose, take the sax back to the rental place and tell them to do a proper overhaul (cleaning the screws and rods so they won't work free, levellng pads, etc). You're paying good money right now for a poor axe.
According to Charles Harris, who has been playing saxophone 35 years and been repairing them since 25 years, he has answered this question in a relative way on qoura.com
You can read the whole thing here:
Hope this helps!