Pay attention to the article itself. You can often find the copyright date in small print at the bottom of the page. It can also help to read through the article. You may be able to estimate the date it was published just by the references it makes.
Find a physical copy of the journal that the article was published in. Search for it by using the catalog in a local library. A university library will likely carry a wider selection of journals than public libraries.
Search for the article on the Google Scholar website, listed as a resource below. You can search by title, author or keyword, and the date the article was published will appear beneath the title in the search results.
Check if the author of the article has a website. Many of these websites have a list of an author's published work, often with the dates included.