After you run the use case, open the developer tools. Then try to find out the <span>, div or <p>, in which this text area is present. You will see element.stye. In that try to put "word-break: break-word;" and observe whether the issue gets rectified.
You should be able to debug this using these steps. Also, please let us know if this issue is browser specific.
Another option is to created a new text input format in the skin and add Attribute in Additional styles of text in the attribute add word-break :break-all. Using this will help in avoiding inline styles and also can be reused.