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1. To make the crust, preheat the oven to 350u00b0F. Stir together the cracker crumbs, brown sugar, flour, and salt in a medium bowl. Add the butter and toss with a fork until well combined. If the mixture is too dry, add water 1 tablespoon at a time. Press the crumb mixture evenly over the bottom of a 10-inch springform pan. Bake for about 15 minutes, until golden brown around the edges. Remove from the oven and let cool completely.
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2. To make the filling, reduce the oven temperature to 325u00b0F. Place one oven rack on the lowest rack and one in the middle of the oven. Beat the cream cheese, yogurt, and peanut butter in the bowl of an electric mixer on medium speed for about 3 minutes, until smooth. Add the sugar, vanilla extract, and salt and beat for another minute, until fully incorporated. Add the eggs, one at a time, blending well after each addition.
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3. Pour the filling into the cooled crust. Fill a large roasting pan halfway with hot water and place on the lowest oven rack to create steam. Place the cheesecake on the center rack and bake for about 1 hour and 40 minutes, until the cheesecake is set around the edges and jiggles only slightly when gently shaken. Remove from the oven and chill in the refrigerator for at least 8 hours, or overnight.
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4. To make the topping, melt the butter in a medium skillet over medium-high heat. Add the sugar and water and stir for a few seconds, until the sugar dissolves. Add the bananas and cook for 3 to 4 minutes, stirring occasionally, until the bananas are tender and lightly caramelized.
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5. To serve, cut the cheesecake into wedges and spoon the caramelized bananas over the top.
