Using \textsuperscript to put commas between consecutive footnotes in amsart document

Seems, as David Carlisle said, like a bug in amsart. You can use the following definition of \@makefnmark in the preamble of your document to get the smaller size in the document:

\makeatletter
\def\@makefnmark{%
  \leavevmode
  \raise.9ex\hbox{\fontsize\sf@size\z@\normalfont\tiny\@thefnmark}}
\makeatother

A complete example:

\documentclass{amsart}

\makeatletter
\def\@makefnmark{%
  \leavevmode
  \raise.9ex\hbox{\fontsize\sf@size\z@\normalfont\tiny\@thefnmark}}
\makeatother

\begin{document}
No comma.\footnote{One.}\footnote{Two.} 
Comma, without font enlargement.\footnote{One.}\textsuperscript{,}\footnote{Two.}
\end{document}

I think this could count as a bug in amsart, but it looks like the first time math is used the math font sizes are readjusted. Footnote markers change from 6pt to 7pt. If you add some mathematics at the beginning of the document, even hidden such as

\setbox0\hbox{$s$}

then a consistent (the larger) size is used throughout.

This seems like a LaTeX bug not ams bug. Try

\documentclass{article}
\title{A Document}
\author{Author}
\begin{document}
\footnote{Test}
\makeatletter
\showthe\sf@size
\makeatother
$1+2=3$
\footnote{Test}
\end{document}

which gives

\sf@size ->6

but

\documentclass{article}
\title{A Document}
\author{Author}
\begin{document}
\footnote{Test}
$1+2=3$
\makeatletter
\showthe\sf@size
\makeatother
\footnote{Test}
\end{document}

gives

\sf@size ->7

That is before math mode, \sf@size=6 but after math mode \sf@size=7 (due to LaTeX redefining \sf@size i.e. \DeclareMathSizes{\@xpt}{\@xpt}{7}{5}).

Let's now look at the difference between \@makefnmark definition in LaTeX and ams. In file latex.ltx, \@makefnmark is defined as

\def\@makefnmark{\hbox{\@textsuperscript{\normalfont\@thefnmark}}}

That is you enter math mode and insert footnote number so the value of \sf@size for each footnote is 7.

but in ams classes, this is how \@makefnmark is defined as

\def\@makefnmark{%
  \leavevmode
  \raise.9ex\hbox{\fontsize\sf@size\z@\normalfont\@thefnmark}%
}

That is footnote number is always, entered in horizontal mode (you never enter math mode). So if you are wondering what happens in

\documentclass{article}
\title{A Document}
\author{Author}
\begin{document}
\footnote{Test}
$1+2=3$
\makeatletter
\showthe\sf@size
\makeatother
\footnote{Test}
\end{document}

is that for the first footnote \sf@size=6 but once LaTeX sees the math mode, it changes \sf@size=7 so the next footnote number is typeset with \sf@size=7. In LaTeX, since you always typeset the footnote number in math mode, then \sf@size=7 for all footnotes but in ams classes, since footnote number is typeset in horizontal mode, the value of \sf@size changes from 6 to 7 once you typeset some mathematics.