Somehow this question came back up after many years.  I didn't think the answers really got to the essence of the situation.

Asking for a fine moduli space is asking for a several different things 
at once that are related but not quite the same. So not only are the nontrivial automorphisms a problem but also that isomorphisms between bundles can degenerate.  I'll leave it to a licensed Algebraic Geometer to fill out the requirements for being a fine moduli space.  I think it is 
instructive to explain that even in the simplest nontrivial example,
rank 2 bundles on $\mathbb{P}^1$ the situation is already subtle.  A key phenomenon is that
a naive moduli space (isomorphism classes of bundles with a topology that forces holomolorphic (or algebraic) families of bundles is to give rise to a continuous maps to iso classes) typically is not Hausdorff. To see this 
consider rank two bundles on $\mathbb{P}^1$ (working over $\mathbb{C}$).
Suppose further that the degree of the bundle is zero. By the Dedekind-Weber, del Pezzo-Bertini, Birkhoff, Grothendieck theorem such bundle
is isomorphic to a sum $\cal{O}(d)\oplus \cal{O}(-d)$ (say $d\ge 0$ to rid ourselves of some ambiguity).  So you great the set of isomorphism 
classes is parametrized by $\mathbb{Z}_{\ge 0}$.  Viewing $\mathbb{P}^1$
as $\mathbb{C}\cup \infty$ with coordinate $z$ around $0$ and $1/z$ around infinity you can write 
the transition function for such bundle as 

\begin{pmatrix} z^{-d} & 0 \\ 0 & z^{d}
\end{pmatrix} 

which admits the deformation
\begin{equation}g_{d,t}=\begin{pmatrix} z^{-d} & 0 \\ t & z^{d}
\end{pmatrix}.\end{equation}


Define a new trivialization (when $t\ne 0$) away from $\infty$ by 
\begin{equation}
\{\begin{pmatrix} z^d\\-t
\end{pmatrix}, 
\begin{pmatrix} 1 \\0\end{pmatrix}\}.
\end{equation}
This gets mapped under $g_{d,t}$ to 
\begin{equation}
\{\begin{pmatrix}1 \\ 0 \end{pmatrix},\begin{pmatrix} z^{-d} \\ t
\end{pmatrix}\}.
\end{equation}
This is trivialization that extends over $\infty$ (when $t\ne 0$). Thus the family of bundles with transition function $g_{d,t}$ gives the trivial bundle for $t\ne 0$ and $\cal{O}(d)\oplus \cal{O}(-d)$ for $t=0$.  This
shows that the closure of the trivial bundle contains every bundle. More generally the closure of $\cal{O}(d)\oplus \cal{O}(-d)$ contains $\cal{O}(e)\oplus \cal{O}(-e)$ for all $e \ge d$.