Minipage and multicols, vertical alignment

You have to set an anchor at the top for [t] to take effect, because the top line inside the multicols environment is not the top line in the enveloping minipage: \vspace{0pt} is sufficient. Also you need to countermand the insertion of a glob of glue (we need to back up by the difference between \topskip and the height of a strut) and to set an initial \strut to ensure good alignment between the baselines.

\documentclass{article}

\usepackage{multicol,calc}
\usepackage{blindtext}

\begin{document}

\noindent % or the indent would apply
\begin{minipage}[t]{0.5\linewidth-0.5em} % some separation
  \vspace{0pt} % anchor for [t]
  \vspace{\dimexpr\ht\strutbox-\topskip}% remove excess glue
  \begin{multicols}{2}
    \strut\makebox[0pt][l]{\vrule width 10cm height 0pt depth 0.1pt}%
    \blindtext
  \end{multicols}
\end{minipage}\hfill
\begin{minipage}[t]{0.5\linewidth-0.5em} % some separation
  \vspace{0pt} % anchor for [t]
  \strut\blindtext
\end{minipage}

\end{document}

The rule shows the alignment. Thanks to Frank Mittelbach for spotting a mistake.

Wouldn't using paracol and multicol be an easier approach?

\documentclass{article}
\usepackage{paracol}
\usepackage{multicol}
\usepackage{mwe}

\begin{document}
   \begin{paracol}{2}
       \begin{multicols}{2}
            \blindtext
        \end{multicols}
  \switchcolumn
     \blindtext
  \end{paracol}
\end{document}