Preheat the oven to 350°F (175°C). Grease a 9x5 inch loaf pan.
In a mixing bowl, mash the ripe bananas with a fork or potato masher until smooth.
Stir in the melted butter.
Mix in the baking soda and salt.
Stir in the sugar, beaten eggs, and vanilla extract.
Mix in the flour until just incorporated.
Fold in the chopped walnuts.
Pour the batter into the prepared loaf pan.
Bake in preheated oven for 60-75 minutes, or until a toothpick inserted into the center comes out clean.
Cool bread in pan for 10 minutes, then turn out onto a wire rack to cool completely.