documentclass{article}

begin{document}

section{Start}
Start text
section{Middle}
Middle text
begin{table}[h]
    centering
    begin{tabular}{l}
    first table\
    end{tabular}
caption{A normal table}
end{table}

section*{Supplemental}

begin{table}[h]
    centering
    begin{tabular}{l}
        table in supplemental part\
    end{tabular}
caption{This should read: ``Table S1: caption text''}
end{table}

end{document}

documentclass{article}
usepackage{titlesec}% http://ctan.org/pkg/titlesec
newcounter{mytable}newcounter{myfigure}
titleformat{section}% <command>
  [hang]% <shape>
  {setcounter{mytable}{value{table}}setcounter{myfigure}{value{figure}}% Store table/figure counters
   setcounter{table}{0}globaldefthetable{Sarabic{table}}%
   setcounter{figure}{0}globaldefthefigure{Sarabic{figure}}%
   normalfontLargebfseries}% <format>
  {setcounter{table}{value{mytable}}setcounter{figure}{value{myfigure}}% Restore table/figure counters
   globaldefthetable{arabic{table}}%
   globaldefthefigure{arabic{figure}}%
   thesection}% <label>
  {2.3ex plus.2ex}% <sep>
  {}% <before code>
  %[]% <after code>
begin{document}

tableofcontents
listoftables

section{Start}
Start text

section{Middle}
Middle text
begin{table}[ht]
    centering STUFF
caption{A normal table}
end{table}

begin{table}[ht]
    centering STUFF
caption{A normal table}
end{table}

section*{Supplemental}

begin{table}[ht]
    centering SUPPLEMENTAL STUFF
caption{A supplemental table}
end{table}

begin{table}[ht]
    centering SUPPLEMENTAL STUFF
caption{A supplemental table}
end{table}

section{Last}
End text
begin{table}[ht]
    centering STUFF
caption{A normal table}
end{table}

begin{table}[ht]
    centering STUFF
caption{A normal table}
end{table}

section*{Supplemental}

begin{table}[ht]
    centering SUPPLEMENTAL STUFF
caption{A supplemental table}
end{table}

begin{table}[ht]
    centering SUPPLEMENTAL STUFF
caption{A supplemental table}
end{table}

end{document}

newcommandstartsupplement{%
    makeatletter
       setcounter{table}{0}
       renewcommand{thetable}{Sarabicc@table}
       setcounter{figure}{0}
       renewcommand{thefigure}{S@arabicc@figure}
    makeatother}