Marking white borders with black marker
Yes, but only if the head judge agrees. The Mtg Tournament rules require that the card has white or black borders (no UN cards or World Champion cards), and the alteration cannot make the card marked. As ghoppe points out, you might have to worry about bleed unless you use Opaque card sleeves. If you choose to do this, a judge will likely still be called to verify that the cards aren't marked for randomization purposes if sleeves aren't used, and the cards are legal white\black bordered cards and not World Champion cards in either case.