Describe the action in your writing instead of explaining it. This is referred to among writers as showing vs. telling. For example, instead of writing "Ella was mad," write "Ella balled her fists and gritted her teeth. A surge shot through her body and she questioned if she truly believed it was wrong to punch someone in the face." Showing allows you to craft scenes of varying levels of emotion and exert tighter control over the overall emotional landscape.
Write in third-person point of view instead of first-person point of view. Third person creates distance between the reader and the narrator by relaying information from a removed perspective. First person directly engages the reader with the emotional core of the piece by saying "I feel" or "I want."
Engage all the senses to set the scene. Describe what your readers can see, smell, taste, hear and touch instead of just what they feel. Doing so will help ground readers in the moment rather than overwhelm them with the negative elements of the situation.
Give your reader breaks after especially intense moments. Break up the piece with transitional paragraphs about your character or subjects' lives or the situation you're covering to diffuse intense moments. Add back-story or research to articles and stories or switch to more positive news in a letter.
Include a subplot in fiction writing that explores the silver lining in some of your negative or dark themes. The human experience is made up of both adversity and triumph. At the very least, negative experiences teach us lessons about ourselves or expose a greater truth about humanity. Journalistic pieces, letters and other works can also contain underlying themes of hope, learning or optimism.
Explore the subject or situation from multiple perspectives and angles try to find other ways to describe it. Interview experts or talk to people with experiences that relate to the themes or subjects you're writing about to help broaden your understanding.
Vary your vocabulary to make sure you're not repeating common negative words and phrases. Not only will varying your vocabulary and sentence structures make your writing sound less negative, it will also make it sound less repetitive.
Use "I" statements rather than accusing others or making judgements, especially in a letter, email and online writing. For example, instead of saying, "You made me mad when you carelessly lost my jacket," say, "I felt upset because I really liked that jacket."
Trust your readers or audience to come to the right conclusions on their own. If you write well, the reader will understand what's happening, connect with the emotional core of the piece and follow what you're trying to convey without you having to remind them about how you'd like them to feel.