Can negative toughness effects kill a creature?
A creature with 0 or less toughness will be sent to the graveyard immediately, so yes, getting -3/-3 can kill a creature.

Note that this is not technically being "destroyed", it is a different way of sending the creature to the graveyard. This matters for things like regeneration.

From the basic MTG rules:


  Your creatures go to your graveyard if the damage they’re dealt 
  in a single turn is equal to or greater than their toughness, 
  or if their toughness is reduced to 0 or less.