I can't see a problem in the selected parts you have shown here.
Here is a working file that has numeric footnotes:
Code:
\documentclass[english]{article}
\usepackage{times}
\usepackage{babel}
\begin{document}
\renewcommand{\thefootnote}{\arabic{footnote}}
Here is a footnote.\footnote{example footnote}
\end{document}
In your file, is there something before the renewcommand that is causing it to be missed (though then I would expect the default to be numeric)? Or is there another renewcommand that is overriding it?