I'm not so much asking about the veracity of the story here; I understand that we have virtually no records about it and aren't even sure of the identity of the mansa in question. However, I've had a hard time learning much about any West African ships of the time beyond river canoes that seem ill-suited to even think of going out in to open ocean (though that's "seem ill-suited" to my eye, which is very much an ignorant one on the matter of sailing). If any exploration of the Atlantic was conducted by Mali around this time, what would the ships be like?
The Mali Empire was primarily concentrated along the highlands of Mandeland (located in contemporary Republic of Guinea and southern Republic of Mali) and along the Niger River Valley, with its downstream border being somewhere between the cities of Kawkaw (modern Gao) and Kukiya (modern Bentiya). At its heyday, the Mali reached the distant lands of some saharan cities (like Walata, Essouk/Tadmakka and Takedda) all the way to the Atlantic Coast, between the Senegal and Gambia rivers.
With that data in mind, we can proceed to investigate the watercourses the Malians would be familiar with and figure out the boats they used to navigate them. Namely, they were familiar with the slow waters of major rivers (Niger, Senegal and Gambia) and with the coast of modern day Senegambia. Sadly, we lack direct sources about what types of boats were available at that time (14th century) in that particular area (River valleys and Senegambian coast). However, we do have a lot of accounts from european traders from the 16th century onwards, and those are quite useful.
European traders, sailing along the Senegambian coast, quickly found africans (wolofs, serers, mandinkas, bizagos and others) using canoes - some quite large, that could transport cattle - to navigate the ocean. Those canoes lacked sails and depended on the strength of rowers to get somewhere. The fact that the Cape Verde islands (about 600 kilometers from the coast) were uninhabited prior to the arrival of european sailors is strong evidence that the local african canoes didn't get very far into the open sea, prefering coastal areas.
Those canoes were mostly made out a single tree and quite effective at their purpose (i.e. navigate around the coast and into river estuaries), but sometimes more diverse materials would be used.
In the larger and calmer rivers inland, canoes were used as well, but also bigger boats made out of several pieces of wood, capable of carrying lots of people and animals. They, however, also lacked proper sails and relied on people (either rowing or pushing against the riverbed with paddles) or simply the gentle currents of the river.
TL;DR: Basically canoes and boats that lacked sails and, despite their occasional large size, lacked the proper resistance to sustain damage caused by waves or fight sea currents. They wouldn't get far and, in the odd occasion that they did, the boats would be at the mercy of the ocean.
This has been asked multiple times, searching for 'mansa musa atlantic' yields for example this topic with a nice answer by /u/MansaMontezuma