Short answer:

Sailors commonly used dogs rather than cats to control rodents.

Discussion:

More robust animals, killed more. Cats aren't really fit for purpose when it comes to rats on a sailing ship. They'll kill a mouse or three, but they won't keep on going. Rats also got big and nasty living on ships, big enough that they were too much for a cat to handle. Dogs are more robust, tougher in a fight, and were preferred by sailors. They also did extra duty as guard dogs, not something that cats do . . . (you'll sometimes hear it suggested that there's a connection to the naval "dog watch", but there's no evidence for that).

There are more than a dozen dog breeds developed fo rat catching, including terriers and pinschers, and there are even dogs bred specifically as ratters for ships and boats, notably Schipperkes.

So, for example, on the wreck of the Mary Rose, a carrack in the Navy of Henry VIII (16th century) there was found the skeleton of a dog, presumably the ship's dog, and named by marine archaeologists "Hatch", with a terrier type phsyiogonomy. His (its a "he") skeleton is on display at the Mary Rose Museum in Portsmouth.

We have plenty of references to dogs on early sailing vessels, including more than a few references to war dogs on Spanish vessels. I can't find similar references to cats. We've also got history of beloved dogs being given honorary ranks in navies -- Bamse, a St Bernard in the Royal Norwegian Navy and "Just Nuisance" a Great Dane in the Royal Navy (British), both during the Second World War.

Dogs aren't picky eaters.

See:

Zouganelis, Ogden et al. "An old dog and new tricks: Genetic analysis of a Tudor dog recovered from the Mary Rose wreck". Forensic Science International, Volume 245, 2014, pp 51-57,