How to add extra space between the left-side legend and the left parenthesis of a matrix in the presence of a vertical line?

Insert a space within the column specification of your blockarray:

\documentclass{article}

\usepackage{booktabs}
\usepackage{blkarray}

\begin{document}

\[
  \mathbf{R} =
  \begin{blockarray}{ cccccc }
    & \lambda_1 & \lambda_2 & \lambda_3 & \lambda_4 \\
    \cmidrule{2-6}
    \begin{block}{@{} c @{\hspace{1ex}} | @{\quad} (ccccc) }
      \tau_1 &           -1 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\
      \tau_2 & \phantom{-}0 &           -1 & \phantom{-}1 & \phantom{-}0 \\
      \tau_3 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 &           -1 \\
      \tau_4 &           -1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 \\
    \end{block}
  \end{blockarray}
\]

\end{document}

Also note the removal of the inter-column space (using @{}) at the beginning of the nested block.

With {pNiceArray} of nicematrix.

\documentclass{article}
\usepackage{nicematrix,tikz}

\begin{document}

\[\mathbf{R}=
\begin{pNiceArray}{cccc}[first-col,first-row]
      & \lambda_1 & \lambda_2 & \lambda_3 & \lambda_4 \\
\hline 
\noalign{\vskip2pt}
  \tau_1 &           -1 & \phantom{-}1 & \phantom{-}0 & \phantom{-}0 \\
  \tau_2 & \phantom{-}0 &           -1 & \phantom{-}1 & \phantom{-}0 \\
  \tau_3 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 &           -1 \\
  \tau_4 &           -1 & \phantom{-}0 & \phantom{-}0 & \phantom{-}1 \\
\CodeAfter
\tikz \draw ([xshift=-2pt]row-1-|col-1) -- ([xshift=-2pt]row-5-|col-1) ;
\end{pNiceArray}\]

\end{document}