How many adjacent intersections per harbor are available for trading in Catan?
Each harbor provides exactly 2 spaces where a settlement can be built to gain access to the harbor. Because they are adjacent, only 1 of the 2 spaces in each harbor can ever be used in the same game.

The image provided by Acccumulation is very helpful, and gives the correct answer. However, I would like to provide an image and rules quote from the third edition of the game, because I believe this clears up any possible ambiguity. To know that the image in the other answer is correct, you have to know that a "harbor" is actually one of the stone jetties drawn on the border; not one of the ships drawn on the border.



In this edition (which uses the same rules for harbors as the newest editions), it is clearer that each harbor has 2 spaces; because of the 2 dotted lines connecting the resource icon to the building intersections.

Also, in the setup rules for placing the harbors, it says:


  
  Then, place harbor hexes in the open coastal spaces between the sea hexes. Each harbor hex should face the island, so that the 2 harbor corners (denoted by doted half-circles), align with the longest row of adjoining terrain hexes.
  


This rule clarifies that the rotation of the harbor tiles is important, because only 2 of the corners of the hex actually count as harbor spaces.

The newer editions replaced these water and harbor hexes with a border, but the rules for harbors stayed the same.