The second line is not part of the title, but of the byline; title style does not apply. Microsoft Word's "grammar" checker is, like all similar machine solutions, primitive; it does not detect that you are not starting a sentence.
Whether or not to capitalize the by in the byline, or omit it altogether, and in what contexts (e.g. research paper cover page as opposed to a magazine article), depends on the style guide you are using. There is no single convention to strive for other than consistency.
For what it's worth, The New York Times always capitalizes by, as does the Los Angeles Times, the Washington Post, and The Wall Street Journal.