Can you safely shower and swim with a stainless Rolex watch (1970's
Air-King) that has been kept in good repair?

open the back case and verify the seal is still intact. If the small
rubber like oring is still present then the watch will withstand water
from a shower. Waterproof? I do not believe it was designed to be
submeged to any depth greater than 1ft.

May the Lord lead you, and you be wise enough to follow.

albsure-ga,

To explain in a little more detail, virtually no vintage watch should
be considered waterproof or water-resistant to any great degree
without specific maintenance.  Water has 3 main points of entry:

1) Under the winding crown
2) Around the crystal edge
3) Through the caseback (in screwback or snapback cases)

The winding crown and caseback usually have one or more sets of rubber
seals installed at the factory to keep out water.  Unfortunately,
these seals will age and crack over time and will lose their ability
to keep out water.  The crown o-rings or seals will wear each time you
manually wind the watch and each time you pull the crown out to set
the time.  Eventually, it loses the ability to keep out water and
needs to be replaced.  In addition, if you've ever had a repair shop
unscrew/unsnap the caseback of your watch and then put it back on
without replacing the o-ring seal, there is a good chance that it is
no longer water-resistant to any great degree.

If you really need to be sure that your watch is waterproof, have all
the rubber seals replaced and take it to a watchmaker that has one of
the depth-testing devices in his/her arsenal of tools.  I believe that
Rolex mandates that its authorized dealers have this piece of
equipment in their shop.

Good luck!

Greg