The following adjustment to \@makecaption
places the figure caption inside a box that adjusts to the width of the caption:

\documentclass{article}
\usepackage{graphicx}% http://ctan.org/pkg/graphicx
\makeatletter
\long\def\@makecaption#1#2{%
\vskip\abovecaptionskip
\sbox\@tempboxa{\fbox{#1: #2}}%
\ifdim \wd\@tempboxa >\hsize
\fbox{\parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{#1: #2}}\par
\else
\global \@minipagefalse
\hb@xt@\hsize{\hfil\box\@tempboxa\hfil}%
\fi
\vskip\belowcaptionskip}
\makeatother
\begin{document}
\begin{figure}
\centering
\includegraphics[width=100pt]{example-image}
\caption{I sure hope I can be boxed separate from the picture\ldots}
\caption{I sure hope I can be boxed separate from the picture.
I sure hope I can be boxed separate from the picture.
I sure hope I can be boxed separate from the picture.}
\end{figure}
\end{document}
The definition of \@makecaption
was taken (and changed) from article.cls
.
caption
also allows for setting a caption style. So, if you're interested in a fixed-width caption, you can use something similar to the following:

\documentclass{article}
\usepackage{caption,graphicx}% http://ctan.org/pkg/{caption,graphicx}
\DeclareCaptionFormat{plain}{%
\fbox{\parbox{\dimexpr\linewidth-2\fboxsep-2\fboxrule}{\centering #1#2#3}}}
\begin{document}
\begin{figure}
\centering
\includegraphics[width=100pt]{example-image}
\caption{I sure hope I can be boxed separate from the picture\ldots}
\caption{I sure hope I can be boxed separate from the picture.
I sure hope I can be boxed separate from the picture.
I sure hope I can be boxed separate from the picture.}
\end{figure}
\end{document}
The style is set to \centering #1#2#3
where #1
refers to the caption label, #2
the caption label separator and #3
the caption text. See section 4 Own enhancements (p 24) of the caption
package documentation.
\caption
.