A classic inspired by The Joy of Cooking. There is no better banana bread, in my opinion.
Preheat oven to 350° and position rack in lower third of oven. Butter a 9x5 loaf pan and set aside.
Mash really ripe bananas with a fork and set aside.
Combine the flour, salt, baking soda, and baking powder in a small container and whisk or sift together thoroughly.
In a mixer bowl, beat the softened butter, sugar, and vanilla on high speed until the mixtures is light in color and fluffy, about 3-4 minutes.
Pour the flour mixture into the butter/sugar bowl in three parts, blending on low speed until just combined with each addition. When the flour is entirely incorporated the mixture will look dry and crumbly. Be careful not to over mix.
Lightly beat the eggs in a separate dish and then add to the mixer bowl. Mix on medium speed until just combined.
Gently fold in the mashed bananas and walnuts until barely combined.
Pour the batter into the pan and lightly tap on the counter to even the surface and eliminate air bubbles. Bake the loaf for 50-60 minutes or until a toothpick inserted in the center comes out clean. Let cool in the pan on a rack for 5-10 minutes before unmolding to cool completely.