Visit the office of the newspaper or magazine which published the article. Newspapers and magazines keep archives and may let you access it. Telephone their office before you visit.
Go to the publisher's website and find the "archives" section. Established newspapers like "The New York Times" let you access old articles for a fee while others let you access the archives for free.
Access Google News archive. Click on "Advanced archive search" and enter the date on the field next to "Return results published between" text. Enter exact date in mm/dd/yyyy format. Click on "Search Archive" button to find articles written on that day.
Go to Newspaper Archive, and search for the article you are interested in. Newspaper Archive claims to hold digital copies of newspaper starting with the year 1600. Newspaper Archive offers an annual, semi-annual or 3-month subscription plan. Choose according to your needs.
Visit your nearest public library, or college library if you are a college student. Ask the librarian if they maintain physical archives, microfilms or a database. Most libraries also allow access to online databases, such as Lexis Nexis, that let you search for articles published on a particular day.