Yes, some monkeys do peel bananas from the floral end, if they get access to them, but not all the time.
While Sklivvz's answer is correct about the lack of access of monkeys to bananas in the wild, monkeys who live next to humans have access to bananas, and, at least some of them, do peel the banana from the floral side.
Here are some videos depicting monkeys peeling a banana from the floral side. Please take note, however, that the monkeys don't do it as "neatly" as it's depicted in the video by the human, and use their teeth to help them peel the banana.
But this is not true for all monkeys or apes.
Some apes don't even bother to peel their bananas, as is depicted in the video Orangutans eating bananas at feeding station in Borneo, where several Orangutans are seen just putting one or several whole bananas in their mouth and eating it as is, with the peel.
Also, a note about the wording of that particular claim, there is no "right" way to peel a banana, while the claim says that peeling from the floral end is the "right" way, what they actually mean is that it's an easier way, which is subjective.