The US Census Bureau declared the end of the frontier in 1890 because there were no significant unclaimed/unsettled lands and there was no longer a significant line separating settled and unsettled land. What might have gone as an unnoticed declaration from an insignificant agency became the cornerstone of an important speech in 1893 by historian Frederick Jackson Turner, who suggested that the existence of a frontier was an important factor not only in the expansion and character of the West but also in the very nature of American culture and government in general. His "Turnerian thesis" became a matter of debate for the next 90+ years; and I think it is fair to say that western historians were glad to bury the discussion and move on to other things after all those decades.

The end of the frontier does not necessarily mark the end of the "Wild West," but the problem with that term is that it is more useful in media - dime novels and later movies - than it is in the academic study of the West (unless it is the history of that media). The Wild West is a fantasy that people use to romanticize about the region. In its true home - as a term to describe how the West is viewed in some media - the Wild West is still alive and well.