I've studied transoceanic botanical exchanges of other plants, but to my knowledge, there hasn't been a book or article length study on the banana in the early modern Atlantic. It would actually be a great project. I am also unaware of any studies that explore Indigenous encounters/adaptations of the banana in the Americas. My guess is that widespread cultivation among Indigenous people occurred in the nineteenth century in Central America. That's just a hypothesis though. I recommend checking out John Soluri's Banana Cultures: Agriculture, Consumption, and Environmental Change in Honduras and the United States as a starting point for modern Latin America banana history.

Your question regards the early modern world. The banana is native to Southeast Asia. It was widely cultivated throughout the Eastern Hemisphere tropics by the early modern period and may have already reached the Americas via Polynesian contact with the Americas. If you search for Polynesian contact with South America on AskHistorians, you might find a few posts about possible contact. I am not an expert on that, but from what I can tell, the case for precolumbian Polynesian contact does not at this time use bananas as a line of evidence. Maybe it will emerge in the future!

The cultivation of bananas in the Americas is then related to the Columbian Exchange. A criticism of the older historiography of the Columbian Exchange is that it focused on the fact that crops moved, rather than the specific circumstances and mechanisms of that movement. Many holes remain about why people adopted new plants/animals and what new plants/animals meant to people. On top of that, the emphasis on movement inadvertantly gave most of the agency to Europeans, whose sources were the ones consulted to write those histories. Perhaps unsurprisingly in regards to your question, the common attribution about the banana is that a friar brought the banana to Santo Domingo in 1516, where it took to the tropical climate and fertile soil of the island. Soon, it spread throughout the Caribbean.

The banana's story however appears to be richer. In their book In the Shadow of Africa: Africa's Botanical Legacy in the Atlantic World, Judith Carney and Richard Nicolas Rosomoff contend that it was likely not a friar, but rather Africans who spread the banana widely in the Caribbean. They argue that the expansion of bananas, plantains, and many other African cultivars mirrored the expansion of slavery. Bananas and plantains moved with Africans from the west coast of Africa to Portuguese and Spanish sugar islands in the Atlantic during the fifteenth century, then to the Caribbean with enslaved Africans and European traffickers in the early sixteenth century. Regardless of whether a European was the first person to cultivate them, Carney and Rosomoff reveal that it was Africans who provided the critical mass necessary to spread new crops.

Their evidence is compelling but somewhat circumstantial. This is by necessity because enslaved Africans did not leave many sources in Spanish and Portuguese archives. However, Carney and Rosomoff note that European traffickers generally bought whatever African foodstuffs were available to provision slave ships before setting sail for the Americas. Slavers believed that it was generally better to feed Africans foods that they were accustomed to eating. This had to do with the belief that healthy bodies were made up of a balance of humors, which drastic dietary changes could throw off. Once bananas were planted on plantations or in their own gardens in the Americas, enslaved Africans drew upon their existing agricultural knowledge and skills to grow the bananas and plantains, and also other African crops like millet, rice, yams, and manioc. Further evidence comes from maroon communities, who also widely cultivated well-known and lesser-known African crops to provision their autonomous communities. Today, many of these lesser-known African crops continue grow wild in the Caribbean and Brazil, a direct legacy of Africa's legacy on the Americas. The value and uses of these plants are obscure to most of us, but people with the right knowledge (locals and ethnographers) can spot these plants with ease.

Carney and Rosomoff's theory is compelling, and I find it convincing. It is also part of a growing body of research focused on the African Atlantic. Enslaved Africans were far and away the largest group of people ^([who were forced]) to cross the Atlantic before the nineteenth century, yet their influence is largely absent from scholarship on the Atlantic and colonial Latin America. Kinship ties, languages, religious beliefs, medicine, healing techniques, agricultural techniques, and many other cultural elements crossed the Atlantic with enslaved people. In the case of agriculture and medicine, European "men of science" tended to "discover" "miraculous" medicines, techniques, and methods. They often intentionally erased the African origins of some of their "discoveries." Or they obscured the African-decended people that they learned from.

The banana appears to be a good example of this erasure. If the friar gets credit for growing the first banana, where did he get the seeds? How did he learn the plant was edible? Who grew the banana trees in Santo Domingo and other areas of early cultivation? Who consumed most of the fruits? Who made bananas and plantains into staples of Caribbean diets? It wasn't that friar or a handful of early modern European intellectuals, but rather the millions of enslaved Africans whose unrecorded use/consumption of the plant over generations caused bananas, plantains, and many other plants to spread widely.