How do you counter a fuse card?
It is actual still one spell so "both spells" is not the correct term. So yes, both halves would be countered.

Here the full rules of Fuse:


  702.101 a Fuse is a static ability found on some split cards (see rule 708, “Split Cards”) that applies while the card with fuse is in a player’s hand. If a player casts a split card with fuse from his or her hand, the player may choose to cast both halves of that split card. This choice is made before putting the split card with fuse onto the stack. The resulting spell is a fused split spell.
  
  702.101b A fused split spell has two sets of characteristics and one converted mana cost. The converted mana cost of the spell is a number equal to the total amount of mana in its two mana costs, regardless of color.
  
  702.101c The total cost of a fused split spell includes the mana cost of each half. (See rule 601.2e.)
  
  702.101d As a fused split spell resolves, the controller of the spell follows the instructions of the left half and then follows the
  instructions of the right half.