The reason ships are painted red is because of the high copper content in the paint, which makes it toxic to marine life and prevents the build-up of small organisms known as biofouling. There are experiments with anti-fouling coatings going at least as far back as 1625, but these early coatings were mostly if not entirely useless and weren't necessary after the invention of copper sheathing. But the transition to iron hulls in the early 19th century and the subsequent galvanic corrosion problem led to a renewed interest in anti-fouling paint. It took several decades to come up with one that worked, and the first effective anti-fouling paint was finally introduced in 1860 and its use became widespread after that.

As an aside, red may be the most common color for anti-fouling paint, but it's not the only one. Green has also been a common color (though nowhere near as common as red), and was the color of most anti-fouling paints used by the Italian and Austro-Hungarian navies during the 20th century. Technically you can make it in any color, but that requires adding additional pigments and makes the paint more expensive, so most people go with the cheaper red option.