I can't say what species of grass that is from the photos, but probably it's either hybrid bermudagrass or seashore paspalum, given it's in Mexico. My first guess is bermuda, since the course opened in '99, meaning it was sown in '98, and there wasn't that much paspalum being planted back then.
Paspalum, once established, will tolerate quite a lot of salt spray. The new varieties will provide first-rate, fast and true putting surfaces. It has the tremendous advantage of being one, single variety that can be used to grass the entire course, so it eliminates any cross-contamination, like with bermuda when a 419 fairway variety invades dwarf variety greens. On the down-side, paspalum is a finicky feeder, and the nutrient balance must be kept just exaclty so. It is also more prone to diseases than bermuda.
But bermuda is tough, too, and resists salt better than most other turfgrasses.
Bentgrass and Poa, found in northern climes like Cleveland, are highly susceptible to salt damage.
The main concern with salt affected sites like that is that you keep them flushed as much as possible with fresh water. Don't let the salts build up in the root zone, they need to be kept passing through. It helps a lot if you've got some light sandy soils underneath.
There are other tricks like calcium and gypsum application, to displace the sodium, and acid injections and sulfur burners, to lower the pH and counter harmful bicarbonates.
Dr.'s Duncan and Carrow (forgot the first names) out of the University of Georgia, are the world's leading authorities on salt affected turfgrass sites and seashore paspalum. Anyone interested could try googling their information to learn more.