From the command line (> represents the prompt) you can issue
> latexdef -s fnsymbol
% latex.ltx, line 2234:
\def\fnsymbol#1{\expandafter\@fnsymbol\csname c@#1\endcsname}
OK, we need to know what \@fnsymbol does:
> latexdef -s @fnsymbol
% latex.ltx, line 2254:
\def\@fnsymbol#1{%
\ifcase#1\or \TextOrMath\textasteriskcentered *\or
\TextOrMath \textdagger \dagger\or
\TextOrMath \textdaggerdbl \ddagger \or
\TextOrMath \textsection \mathsection\or
\TextOrMath \textparagraph \mathparagraph\or
\TextOrMath \textbardbl \|\or
\TextOrMath {\textasteriskcentered\textasteriskcentered}{**}\or
\TextOrMath {\textdagger\textdagger}{\dagger\dagger}\or
\TextOrMath {\textdaggerdbl\textdaggerdbl}{\ddagger\ddagger}\else
\@ctrerr \fi
}%
This means that if \fnsymbol appears in text mode, it associates \textdagger for the number 2.
For the specific application you might consider to redefine \@fnsymbol, so as not to redefine \textdagger.
\documentclass{article}
\makeatletter
\let\latex@fnsymbol\@fnsymbol
\renewcommand\@fnsymbol[1]{\ifcase#1\or*\or*\else\@ctrerr\fi}
\newcommand{\restorefnsymbol}{\let\@fnsymbol\latex@fnsymbol}
\makeatother
\setlength{\textheight}{7cm} % just to make a smaller picture
\begin{document}
\title{Test}
\author{Author 1\thanks{Thanks 1} \and
Author 2\thanks{I-should-be-next-to-an-asterisk}}
\maketitle
\restorefnsymbol
Here we have a dagger: $\dagger$
\end{document}
