My Border Terrier definitely does!! Regardless of your Border’s coat colour, you cannot notice the moulting on your floor or carpet. However you will notice his/her fur stuck to your clothes after a cuddling session.
Moulting is a natural process in most dog breeds, nor there is a way to prevent it. Border Terriers can become very hairy, hence the need to groom them regularly. But it is always wise to hand strip them to retain the under-layer coat, which protects them from natural elements. Otherwise, Borders can be kept nice and tidy with regular brushing of their coat in between appointments to the groomer’s.