The way the prefixes are typeset is controlled by the internal command \@character which, by default, sets them in their own line if the drama* environment is used, and sets them in line if the non-starred version of the environment is used; here's the original definition:
\def\@character#1#2{%
\@namedef{#2}{{\namefont #1}\xspace}
\n@me@ppend@nddef{#2}{\@ppendname}{%
\if@drverse
{\speakstab\speaksfont{#1}\speaksdel\par\nobreak\addvspace{-\parskip}}
\else
\item[#1\speaksdel]
\fi}
}
If you want both environments to treat the prefixes in the same way, you can then omit the conditional branching:
\documentclass{memoir}
\usepackage{dramatist}
\makeatletter
\def\@character#1#2{%
\@namedef{#2}{{\namefont #1}\xspace}
\n@me@ppend@nddef{#2}{\@ppendname}{%
\item[#1\speaksdel]
}
}
\makeatother
\begin{document}
\Character[Poe Prose]{POE}{poe}
\Character[Frost Verse]{FROST}{frost}
\begin{drama*}
\frostspeaks Some say the world will end in fire; \\
Some say in ice. \\
From what I've tasted of desire \\
I hold with those who favor fire. \\
But if it had to perish twice, \\
I think I know enough of hate \\
To know that for destruction ice \\
Is also great \\
And would suffice.
\end{drama*}
\begin{drama}
\poespeaks For the most wild, yet most homely narrative which I am about to pen, I neither expect nor solicit belief. Mad indeed would I be to expect it, in a case where my very senses reject their own evidence.
\end{drama}
\end{document}

\documentclass{...}and ending with\end{document}. – jub0bs Apr 08 '13 at 11:02