Cut the mangoes off with a few inches of stem and set them on their side after an hour or more you can remove the stems. another option is to pick the mango and immediately turn them and set them upside down on some newspapers etc.
the best thing is to not let them get dripped on at all, sap on the fruit only a few minutes is enough to stain them permanently, and even if its not that noticeable when the fruit is green, when it ripens the stains will show.
the stain marks wont change fruit flavor, its just about esthetics . now when removing fruit from the tree, if sap drips down onto other fruit, the stains can attract fungus, this is why picking fruit with a piece of stem still attached is best method.