Problem with enumeration using greek letters

The \alph counter style uses the old fashioned Greek alphabetic numbering.

You can define your own \greekalph macro:

\documentclass[12pt]{article}
\usepackage[utf8]{inputenc}
\usepackage[english,greek]{babel}

\usepackage{enumitem}
\usepackage{multicol} % just for the example

\makeatletter
\newcommand{\greekalph}[1]{\expandafter\@greekalph\csname c@#1\endcsname}
\newcommand{\@greekalph}[1]{%
  \ifcase#1\or
  \textalpha\or\textbeta\or\textgamma\or\textdelta\or\textepsilon\or
  \textzeta\or\texteta\or\texttheta\or\textiota\or\textkappa\or
  \textlambda\or\textmu\or\textnu\or\textxi\or\textomicron\or\textpi\or
  \textrho\or\textsigma\or\texttau\or\textupsilon\or\textphi\or
  \textchi\or\textpsi\or\textomega\else\@ctrerr\fi
}
\AddEnumerateCounter{\greekalph}{\@greekalph}{\textomega}
\makeatother

\AtBeginDocument{\renewcommand{\textstigma}{\textsigma\texttau}}% no stigma

\begin{document}

\begin{multicols}{2}
\begin{enumerate}[label=(\let\textdexiakeraia\relax\alph*)]
\item 1
\item 2
\item 3
\item 4
\item 5
\item 6
\item 7
\item 8
\item 9
\item 10
\item 11
\item 12
\item 13
\item 14
\item 15
\end{enumerate}

\begin{enumerate}[label=(\greekalph*)]
\item 1
\item 2
\item 3
\item 4
\item 5
\item 6
\item 7
\item 8
\item 9
\item 10
\item 11
\item 12
\item 13
\item 14
\item 15
\end{enumerate}

\end{multicols}

\end{document}

With moreenum and [label=(\greek*)] you should get what you want:

\documentclass[12pt]{article}
\usepackage{enumerate}
\usepackage{enumitem}
\usepackage{moreenum}
\usepackage[utf8]{inputenc}
\usepackage[english,greek]{babel}

\begin{document}

    \begin{enumerate}[label=(\greek*)]
        \item 1
        \item 2
        \item 3
        \item 4
        \item 5
        \item 6
        \item 7
    \end{enumerate}

\end{document}