I should imagine that there is no single answer to that, as city directories would have been produced at the local level and there would have been no nationwide standard.
Most of the city directories I am referring to are produced by R.L. Polk City Directories, a company that has been in business since 1870. In any given year, they published directories for multiple cities. I highly recommend this source to all of my genealogy friends. I finally found the answer to my question and I am posting it here in case someone else needs this information.
In a 1911 Directory, after the Abbreviation section, it states, “An asterisk (*) before a name denotes colored.”
As I stated previously, sometimes the asterisk is there and sometimes it is not so I assume that it was subjective and up to the enumerator. I do come across that from time to time.
Dear Ms. Carter,
Thank you for posting your request on History Hub!
The date and location often determines what the asterisks meant. In a number of locations, asterisks usually was used to denote race of African American residents. In other cases, it was used to denote a new resident at a given address. Unfortunately, the directories did not include a key to the symbols used.
We hope this information is helpful.
[Information provided by Elizabeth Burnes, Subject Matter Expert]