Create believable characters then place them in jeopardy. Give your characters real lives, with jobs and goals and people that care about them. If nobody in the movie cares about your characters, there is no reason for the audience to. Once the audience cares about a character, you can threaten the audiences' sympathies by putting that character in dangerous situations. The fear that something bad is going to happen to a character is the foundation of all horror scene writing.
Follow basic scene writing techniques to create drama. A scene should begin as close to the immediate action as possible and end before the action is actually over. This leaves the audience engaged in what is going to continually happen in the following scene. Build upon the drama you create by using horror and action genre conventions. In the movie "Aliens" for example, futuristic Marines with guns fight alien monsters. This mixes guns, often associated with action movies, and monsters, often associated with horror films. The popular "Saw" movie franchise uses devices, such as booby traps and gadgets that maim and dismember. Blood and gore are horror movie conventions while gadgets are often associated with action films.
Create tension through point-of-view. Position the camera so the audience sees things through the character's point-of-view. The benefit is that the audience can only know what the character knows. Additionally, this technique places the audience in the center of the action as if everything on screen is happening to them.
Have your characters be chased by something or someone. The chase is a tried and true movie convention that belongs to both the action genre and the horror genre. If your character chases something, the audience should be afraid of what will happen if he or she doesn't catch it. If your character is being chased, the audience should be afraid of what will happen if he or she is caught.
Use monsters to terrorize your characters. Even more than serial-killers and deranged madmen, monsters lend themselves to action. Monsters have the advantage over human characters; usually being bigger, faster, and having exceptional powers. Monsters also always chase people in movies, often tearing up the surroundings and upping the stakes. Using a monster places your character at an immediate disadvantage and gives the audience something to root for.