The most important older word is bashō, 芭蕉. It comes from Chinese, and the first known occurrence of the word in Japanese materials is from ca. 706, in the Nara Ibun historical records.
Bananas are native to the Eastern tropical areas spanning India, Southeast Asia and Australia. However, wild species don't make good edible fruit; they had to be domesticated, and that happened in Papua New Guinea. Meanwhile, non-edible varieties were prized as an ornamental plant in ancient China. The plant was introduced from China into ancient Japan for this purpose; edible bananas only came in much later.
Bananas have various names in Chinese, and one of them was 芭蕉 bājiāo, which entered Japanese along with the plant (and is now pronounced bashō). The -jiāo 蕉 part is a suffix for broadleaf plants in general. I don’t know what’s the etymology of the bā- 芭 part (any help welcome!), but it has the general look and feel of Chinese loanwords to me; it wouldn't surprise me if it came from the name of the plant in some other Asian language.
The most famous Japanese haiku poet is Matsuo Bashō; this is actually his pen-name, derived from a banana plant he had outside of his quasi-hermit's hut.
Other Chinese names which entered Japanese, but are less common, include 甘蕉 gānjiāo (jp. kanshō), “sweet broadleaf”, dated 1712 in Japan; and 香蕉 xiāngjiāo (jp. kōshō), "fragrant broadleaf".
An uncommon, native Japanese word for it is Niwa-mi-gusa, which the Kokugo Daijiten dictionary glosses as 庭忌草 garden-funeral-herb. Apparently there was a Chinese folk belief that planting one in your garden would bring bad luck, and this was transmitted to Japan. Banana plants of the kind brought to Japan flower very rarely, and this fact was seen with some mystique; from which the plant was also called 優曇華 udonge – the Japanese pronunciation of uḍumbara, a kind of fig which, according to Buddhist legend, blossoms only once every 3000 years.
Sources: Japanese Wikipedia, Kokugo Daijiten, Pleco Chinese dictionary.