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.
The book never says if they live or die.
Klaus Fisch died in 1975.
Klaus Holzkamp died in 1995.
Klaus Mollenhauer died in 1998.
Klaus Samelson died in 1980.
Klaus Vogel died in 2007.
Klaus Weiß died in 2000.
Georg Klaus died in 1974.
Klaus Scholtz died in 1982.
Klaus Sunnanå died in 1980.
Klaus Groth died in 1899.