The normal setting for the sizes are decided as follows:
\DeclareMathSizes{5}{5}{5}{5}
\DeclareMathSizes{6}{6}{5}{5}
\DeclareMathSizes{7}{7}{5}{5}
\DeclareMathSizes{8}{8}{6}{5}
\DeclareMathSizes{9}{9}{6}{5}
\DeclareMathSizes{\@xpt}{\@xpt}{7}{5}
\DeclareMathSizes{\@xipt}{\@xipt}{8}{6}
\DeclareMathSizes{\@xiipt}{\@xiipt}{8}{6}
\DeclareMathSizes{\@xivpt}{\@xivpt}{\@xpt}{7}
\DeclareMathSizes{\@xviipt}{\@xviipt}{\@xiipt}{\@xpt}
\DeclareMathSizes{\@xxpt}{\@xxpt}{\@xivpt}{\@xiipt}
\DeclareMathSizes{\@xxvpt}{\@xxvpt}{\@xxpt}{\@xviipt}
This means that when the font size is 10, the normal math is in 10pt size, first level subscripts (and superscripts) 7pt and second level subscripts (and superscripts) 5pt.
You can change this setting in your preamble, for instance
\makeatletter
\DeclareMathSizes{\@xpt}{\@xpt}{6}{5}
\makeatother
so that \[ s_{A} \]
would give

You have to modify all entries relative to the point sizes you're effectively using. It's best to load \usepackage{lmodern}
if you want to give fractional sizes (or the fix-cm
package).
There's another trick worth noting: one can shift a bit down a subscript by inserting an empty superscript. Here's the result of \[s_{A}\,s^{}_{A}\]

And here is what is obtained adding the previous change with \DeclareMathSizes

One can perform this automatically by loading the subdepth
package.