Beat butter and shortening at medium speed until creamy. Gradually add sugar and crushed vanilla wafers, beating well. Add banana and vanilla; mix well with a spoon.
Combine flour, baking soda and salt. Add to previous mixture; mix well. Stir in white chocolate chips.
Drop dough by teaspoonfuls onto ungreased cookie sheets and bake at 350 degrees for 12 minutes or until golden. Cool 1 minute on cookie sheet. Remove to wire racks to cool completely.