One way could be to use the \ifdimcomp{⟨dimen expression⟩}{⟨relation⟩}{⟨dimen expression⟩}{⟨true⟩}{⟨false⟩} macro to test for different paperwidths (and height in case of 169 and 1610):
\documentclass[
aspectratio=169
% aspectratio=43
]{beamer}
\begin{document}
\makeatletter
\ifdimcomp{\beamer@paperwidth}{=}{12.80cm}{43}{}
\ifdimcomp{\beamer@paperwidth}{=}{14.85cm}{141}{}
\ifdimcomp{\beamer@paperwidth}{=}{13.50cm}{32}{}
\ifdimcomp{\beamer@paperwidth}{=}{12.50cm}{54}{}
\ifdimcomp{\beamer@paperwidth}{=}{14.00cm}{149}{}
\ifdimcomp{\beamer@paperwidth}{=}{16.00cm}{
\ifdimcomp{\beamer@paperheight}{=}{10.00cm}{1610}{}
\ifdimcomp{\beamer@paperheight}{=}{9.00cm}{169}{}
}{}
\makeatother
\end{document}
With the current beamer development version (which should be included in beamer v3.70 or newer), one can now simply use the \insertaspectratio macro:
\documentclass[
aspectratio=169
% aspectratio=43
]{beamer}
\begin{document}
\begin{frame}
\insertaspectratio
\end{frame}
\end{document}