What badge do you need for cut in Pokemon Silver?

You don't need a specific badge to use Cut in Pokemon Silver. You can learn Cut from the Move Tutor in Goldenrod City after you've defeated the Gym Leader there, Whitney.

However, you need the Rock Badge from the Gym Leader in Ecruteak City to use Cut outside of battle. The Rock Badge allows you to use HM moves like Cut, Strength, and Flash.