No, Klaus does not die. His siblings stay alive too.
At the end of the "A Series of Unfortunate Events" series by Lemony Snicket, the Baudelaire orphans finally uncover the secrets of VFD, defeat Count Olaf, and find some resolution. The series concludes with the orphans realizing that life is filled with both fortunate and unfortunate events, but they choose to face them together.
Klaus Abramowsky died in 1998.
Fred Klaus died in 2005.
Klaus Störtebeker died in 1401.
Klaus Egge died in 1979.
Klaus Sletten died in 1946.
Klaus Clusius died in 1963.
Klaus Woerner died in 2005.
Klaus Höhne died in 2006.
Klaus Winter died in 2000.
Klaus Holzkamp died in 1995.
Klaus Dittrich died in 2007.