The name etc. are placed in a \begin{minipage}[b]...\end{minipage}
and the information on the right is in a \begin{tabular}[b]..\end{tabular}
. You can remove or change the alignment arguments [b]
with the patching commands from etoolbox
. The following will give vertical centering, which was your preferred option.
Updated for newer versions of moderncv
:

\documentclass{moderncv}
\moderncvstyle{classic}
\name{John}{Doe}
\address{Street St., 167}{8888 Town}{Country}
\phone{+1~(234)~567~890}
\homepage{xxx.ddd.cc.ee}
\usepackage{etoolbox}
\patchcmd{\makecvhead}{{minipage}[b]}{{minipage}[c]}{}{}
\patchcmd{\makecvhead}{{tabular}[b]}{{tabular}[c]}{}{}
\begin{document}
\makecvtitle
\end{document}
Use [t]
arguments aligns at the top, but this with respect to baselines of the first line of each box, so is often not what you expect.
Working a bit harder we get top alignment as follows:

\documentclass{moderncv}
\moderncvstyle{classic}
\name{John}{Doe}
\address{Street St., 167}{8888 Town}{Country}
\phone{+1~(234)~567~890}
\homepage{xxx.ddd.cc.ee}
\usepackage{etoolbox}
\patchcmd{\makecvhead}{{minipage}[b]}{{minipage}[t]}{}{}
\newsavebox{\mytempbox}
\newlength{\mytitleoffset}
\savebox{\mytempbox}{\addressfont S}
\setlength{\mytitleoffset}{\ht\mytempbox}
\patchcmd{\makecvhead}{\raggedleft\fi}{\raggedleft\fi\vspace{-\mytitleoffset}}{}{}
\patchcmd{\makecvhead}{{tabular}[b]}{{tabular}[t]}{}{}
\begin{document}
\makecvtitle
\end{document}
Note that we have measured the height of the largest character S
in the first line of the address. With other material in that block you would have find something appropriate to measure.