Combine sugar, flour, and salt together in a medium saucepan. Add eggs and stir well. Stir in milk, and cook over low heat, stirring constantly.
When the mixture has thickened enough to coat the back of a metal spoon, remove from heat and continue to stir, cooling slightly. Stir in butter and vanilla until smooth.
Layer pudding with bananas and vanilla wafers in a serving dish. Chill at least 1 hour in the refrigerator before serving.