The counter secnumdepth controls the level up to which sectional units will be numbered, so it doen't make much sense to use this counter to number your structure.
One possibility using the titlesec package:
\documentclass{article}
\usepackage{titlesec}
\titleformat{\section}
{\normalfont\Large\bfseries\filcenter}{Problem \# \thesection}
{0em}{}
\begin{document}
\section{}
text
\section{}
text
\section{}
text
\end{document}
Perhaps you could consider defining a dedicated command/environmet for this instead of (ab)using \section?
Here's a better option, using a theorem-like structure defined with the help of the amsthm package; in this way, the counter is provided and increased automatically (feel free to make the adjustments that best suit your needs):
\documentclass{article}
\usepackage{amsthm}
\newtheoremstyle{mystyle}
{\topsep}{\topsep}{\normalfont}{}{\Large\bfseries}{\newline}{0em}
{\hfil\thmname{#1 \#}\thmnumber{ #2}\thmnote{#3}\hfil}
\theoremstyle{mystyle}
\newtheorem{prob}{Problem}
\begin{document}
\begin{prob}
text
\end{prob}
\begin{prob}
text
\end{prob}
\end{document}
As a side note, \bf is an obsolete command (only provided for compatibility); you should use \bfseries instead; additionally \bf and \bfseries are declarations that do not take arguments; they are to be used as in {\bfseries text} (the braces are only to keep the effect local to a group).
secnumdepthis not the counter to use here. – egreg Nov 2 '12 at 13:44articledefault), since you most likely don't want the text in italic. – barbara beeton Nov 2 '12 at 13:56\Largenor\bfare commands that take an argument. Thus, it's not necessary to encase the subsequent material in braces. The scope of these two commands will end automatically when the\end{center}instruction is encountered. – Mico Nov 2 '12 at 15:20