You are most likely to sound rude in writing at two moments: when you are upset, and when your message might upset someone else. When you are upset, step away before you write. When the reader might be upset, choose words that acknowledge their feelings and never blame them.
Sometimes you just know that things will not go the way you want. That happens in writing too. You hesitate just before you hit the Send button in your email app or the Submit button on a reply form, because you feel that things will go downhill the moment you do.
The wrong time to write
Knowing when not to write is critical, especially in business. When things go wrong, it may not end with a little warning: you could lose your clients or your job.
In general, be extra careful not to sound rude when you are upset, and when you think you might upset others. Here is more on each of the two.
When you are upset
Feeling stressed out at the office and thinking about writing in ALL CAPS to make a point? Don't.
Take a deep breath and step away from the keyboard. It is far too easy to write the wrong things when you are angry, and venting your anger only backfires, for you and for the person you are writing to.
Go for a walk to calm down, or do anything else that gets you back on track. Remember that the other person may not even realize they made you angry. Perhaps they were just having a bad day, and it showed.
When you do write, be courteous and use a positive tone. That way, you won't come across as someone looking for a fight. The article on using different tones in writing has more on finding the right one.
Check your message for misspellings, too: the Spell Checker marks them before you send it. Any typo you make can and will be used against you. Not in a court of law, but quite possibly in your supervisor's office.
When you think you might upset others
This one is closely tied to the first. Even when you are not angry at all, you can still come across as rude and condescending. In writing, people can't see your facial expression or hear the tone of your voice.
When you reply to angry customers, for example, avoid any wording that blames the customer. Acknowledge their frustration and let them know you understand how they feel. Assure them that you will look for the best solution.
If you don't have enough information to handle the customer's issue, say so openly. That is better than guessing wildly at what to do. For the basics of a clear, polite message, see how to write a business email.
Keep one more thing in mind: you can't please everyone. Do your best not to offend people, but some people will take offense whatever you write. Don't worry about it too much.