How to stop cases from spreading to multiple lines

I don't find the layout in the original manuscript easy to follow. I'd go with something like the following instead.

\documentclass{article}
\usepackage{amsmath}
\begin{document}
\begin{equation*}
\begin{cases}
d_{00}=0 \\
d_{ij}=\min 
\begin{cases}
d_{i-1,j-1} + (\text{if $a_i = b_j$, then $0$ else $c_C$}) \\
d_{i-1,j}+c_d\\
d_{i,j-1}+c_i
\end{cases}
\text{if $i > 0$ or $j > 0$} 
\end{cases}
\end{equation*}
\end{document}

Here is one option:

\documentclass{article}

\usepackage{amsmath}

\begin{document}

\begin{equation}
  \setlength{\arraycolsep}{0pt}
  \begin{array}{ r l l }
    \raisebox{\dimexpr-.5\normalbaselineskip-.5\jot}[0pt][0pt]
      {$\left\{\begin{array}{l}\mathstrut\\[\jot]\mathstrut\end{array}\right.$}
    & d_{\infty} = 0, \\[\jot]
    & d_{ij} = \min( & d_{i-1,j-1} + (\text{if } a_i = b_j \text{ then } 0 \text{ else } c_C), \\[\jot]
    &                & d_{i-1,j} + c_D, \\[\jot]
    &                & d_{i,j-1} + c_I), \quad i > 0 \text{ or } j > 0.
  \end{array}
\end{equation}

\end{document}

The entire structure is set inside an array, while the "cases brace" is moved into position from the first line.