Most jewelry that turns a wrist green is plated brass. The plating wears through, the brass underneath meets sweat, and the reaction you see is copper oxidising. Stainless steel does not have that problem, because there is no reactive base metal hiding under the finish.
The base metal is the whole story
Stainless steel is an alloy built around chromium, and chromium forms a thin oxide layer on the surface that reseals itself when it is scratched. That is what stainless means. It is not a coating. Scuff it and the protection comes back on its own.
What the plating is for
Gold tone and white gold tone plating on stainless steel is about color, not protection. The steel underneath is already stable, so if the plating wears at a high friction point, say the inside of a ring or the back of a clasp, what shows through is steel rather than something that will react with your skin.
Hypoallergenic in practice
The usual trigger for a jewelry reaction is nickel. Good stainless steel jewelry is low nickel or nickel free, which is why it suits people who cannot wear costume pieces for more than an hour. If you have reacted to jewelry before, this is the property worth checking, more than the color or the price.
Water, sweat and the gym
Stainless steel handles water. Showering in it will not ruin it. What shortens the life of the finish is chemistry and abrasion: chlorine, salt water over long periods, perfume sprayed directly onto the piece, and hairspray drying on a chain. Put jewelry on last, after product and perfume, and the finish lasts noticeably longer.
Keeping it looking new
Warm water, a drop of dish soap, a soft toothbrush for the links, then dry it fully with a cloth. Two minutes, once a month. Skip the ultrasonic cleaners and the abrasive polishes, since those take plating off faster than daily wear does.
How to tell what you own
Stainless is heavier than aluminium, it is not magnetic in most jewelry grades, and it does not tarnish dark the way sterling silver does. If a piece has gone dull and dark rather than simply scratched, it is probably not steel.