Mulberry is either male or female, do if you have a cv. that is fruitless then you have a male plant. It will never fruit.
If you have a female and find the fruit messy there are crop thinning agents ou there that will reduce the crop, not eliminate them.
Being growth regulators, I beilieve they require a licensed applicator to do the treatments.
The windw of oportuninty is very small and I have never treated mulberry, so I do not know the effectivenss.
The most common product is "Florel". NAA is also used as well as some common pesticides in fruit crop control.