Combat training for the Anglo-American sailor of the eighteenth and early nineteenth century in a Navy was rather limited. There were a number of boarding actions during the naval wars of this period, but it didn't necessarily mean your typical sailor would encounter them regularly in their career in the Navy. It depends on the war and their vessel. Imagine being a sailor on one of the long blockades of a French port during the Age of Revolution. One could spend a long time at sea on not see a boarding action. Also, the ultimate goal of ship action was not to board, but to subdue one's enemy. If that could be done without boarding, it was. These ships had many guns (as in cannon - you call cannons "guns" at sea) and they used them. If a opposing vessel could be disabled enough to make resistance futile and so surrender, that was done. Boarding was risky since it required close combat between a lot of men who often had a mix of experience in terms of boarding (having mostly no/little experience would not be surprising at all). Of course you avoid that if you can.

Now, specifically for the training - there was very little of it. No set school of boarding combat training existed, and a basic system of training with cutlasses for the common sailor didn't exist until about the time of the War of 1812. Captains were left up to their own devices for training. Once in a great while you might find the captain who issues cudgels and wooden swords for training in sword - but it was nowhere near the norm. There was firearms practice, often alongside training on the guns. For the guns, dry runs were normal, though experience with firing live rounds in training was limited. Gunpowder cost money, so there was limited amounts of gunpowder allowed for practice in the Navy. Captains would have to purchase their own powder to pursue further practice. So many captains kept their practice to a minimum, though others prided themselves in highly trained combat crews, and achieving short reload times (there is the legendary "3 shots in five minutes" though 2 minutes to the round was considered remarkable, and 3-5 minutes per round was considered pretty typical since maintaining fast reload times would exhaust gun crews quickly enough in a long fight that could last hours).

So you had the rare crew that trained in sword combat, the crews that actually trained well in their gunnery, and those who had experience in combat already. Beyond that, your typical sailor could go some time without close combat experience or training. It was on many occasion that two sailors with no combat experience would encounter each other and do their best. For the sword, maybe the guy would be lucky if he participated in or saw a festival in their home village that featured a cudgel combat competition that involved trying to put a bloody gap of particular size on your opponent's skull - some some sailors with no experience had a tendency to hit for the head if they had a sword. Since it took some skill to use a sword well (rather than using it as a iron rod with a handle), it's no wonder firearms were preferred by many - far less skill to learn how to use one of those and could level the playing field against a skilled opponent.

I learned much of this from the two volume work by William Gilkerson, Boarders Away, a rare work (the only one of it's kind really) that studies mostly munition-grade small arms, explosives, and edged weapons used by sailors in the Atlantic World from the later parts of the sixteenth century to the early nineteenth (though more attention goes to the 1750-1815 era).

Most warships during the Age of Fighting Sail had a master-at-arms, a petty officer or a warrant officer responsible for training the sailors to use small arms and melee weapons. Most sailor would probably have a basic proficiency in using boarding axes, cutlasses, boarding pikes, and cutlasses. So an able seaman c.1800, at least in the Royal Navy, was capable of holding his own in a boarding action or as part of a shore party.