As a teenager in the 1960s, I walked up and down it, on a few occasions - all 898 steps - before NPS closed off the stairs. (Trivia: When it was first opened, only men were allowed on the elevator. Women had to use the stairs. One of the memorial stones on the inside wall was donated by the Vatican with an inscription, and was stolen/ removed by some members of the "know nothing" nativist party).
(I assume you mean the one in D.C. There's another in Baltimore, but it's only 1/3 as high).
Per the Huffington Post,
the Washington Monument was completed on this day, December 6, in 1884!
I wonder why?
Ok, I think I found why women and children had to use the stairs..because the elevator was considered to be too dangerous.
So there you. Cool.