Alki Beach is part of Alki Beach Park, 2.5 mile long beach/walkway area on the Puget Sound. On a nice day, the beach is always packed, so be prepared for that. It is also not a swimming beach. It is a nice place to relax, and the plethora of restaurants nearby make this a good place to spend a day.
In 1907 an amusement park called Luna Park was created at the north end of the beach. It did well until it burned down in 1931. In 1945 the city purchased the land, and in-filled it in 1954. These facts are essentially trivial, but vaguely interesting.
A small replica of the Statue of Liberty was installed in 1952 as part of a national effort to commemorate the 40th anniversary of the Boy Scouts of America. In 2006 the deteriorating statue was recast, and in 2008 a modernized plaza space for the statue was finished. I would suggest coming to the beach for the beach. Going out of your way to see these monuments would probably be disappointing.
Near the north end of the park is a monument to the first European settlement in the Puget Sound area on November 13, 1851. This settlement was later abandoned and a new one was established in what is now Pioneer Square. Embedded in the monument is a rock from Plymouth Rock, MA.