The solution depends on the citation style. For styles like authortitle, verbose and their respective variants (and also for biblatex-juradiss) that execute autocite=footnote, we only need to redefine the underlying \smartcite macro so that it does not add parentheses in footnotes (by replacing \mkbibparens with \textnormal).
\documentclass{article}
\usepackage[style=authortitle]{biblatex}
\DeclareCiteCommand{\smartcite}[\iffootnote\textnormal\mkbibfootnote]
{\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite}}
{\multicitedelim}
{\usebibmacro{postnote}}
\DeclareMultiCiteCommand{\smartcites}
[\iffootnote\textnormal\mkbibfootnote]{\smartcite}{\multicitedelim}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@misc{A01,
author = {Author, A.},
year = {2001},
title = {Alpha},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
\null\vfill% just for the example
Some text \autocite{A01}.
Some text.\footnote{A footnote \autocite{A01}.}
\printbibliography
\end{document}

For styles like authoryear that by default execute autocite=inline (which uses \parencite), we need to declare a new autocite option value (say, inlineplainfootnote) that points to a new underlying macro (say, \mysmartcite) that does what you want (add parentheses in normal text, don't add them in footnotes).
\documentclass{article}
\usepackage[style=authoryear]{biblatex}
\DeclareAutoCiteCommand{inlineplainfootnote}{\mysmartcite}{\mysmartcites}
\DeclareCiteCommand{\mysmartcite}[\iffootnote\textnormal\mkbibparens]
{\usebibmacro{prenote}}
{\usebibmacro{citeindex}%
\usebibmacro{cite}}
{\multicitedelim}
{\usebibmacro{postnote}}
\DeclareMultiCiteCommand{\mysmartcites}
[\iffootnote\textnormal\mkbibparens]{\mysmartcite}{\multicitedelim}
\ExecuteBibliographyOptions{autocite=inlineplainfootnote}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@misc{A01,
author = {Author, A.},
year = {2001},
title = {Alpha},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
\null\vfill% just for the example
Some text \autocite{A01}.
Some text.\footnote{A footnote \autocite{A01}.}
\printbibliography
\end{document}

authoryear,authortitle, ...) do you use? – lockstep Aug 22 '12 at 08:10