New answer
The package cleveref allows one to alias different counters to other ones, and this works locally in groups. So one solution is to add
\crefalias{section}{appendix}
just after \begin{appendices}. You can do this globally using \AtBeginEnvironment from etoolbox.

\documentclass{scrartcl}
\usepackage{lipsum} % for dummy text
\usepackage{etoolbox}
\usepackage{appendix}
\usepackage[capitalize]{cleveref}
\AtBeginEnvironment{appendices}{\crefalias{section}{appendix}}
\begin{document}
\section{The first section}
\label{sec:1}
\lipsum[2]
\section{The second section}
\Cref{sec:1} tells us something, while \cref{app:1} tells us something
else! In \cref{app:2}, we see that this is self-contradictory. All
is fine again in \cref{sec:3}.
\begin{appendices}
\section{The first appendix}
\label{app:1}
\lipsum[2]
\subsection{An appendix subsection}
\label{app:2}
\lipsum[3]
\end{appendices}
\section{The third section}
\label{sec:third-section}
\label{sec:3}
\lipsum[4]
\end{document}
Old answer
The package cleveref provides a mechanism for this. Introduce a new type appsec and set up the label name via \crefname; then in the appendices part use \crefalias to get the section type to point to the type appsec:

\documentclass{scrartcl}
\usepackage{lipsum}
\usepackage{appendix}
\usepackage[capitalize]{cleveref}
\Crefname{appsec}{appendix}{appendices}
\begin{document}
\section{The first section}\label{sec:1}
\lipsum[2]
\section{The second section}
\Cref{sec:1} tells us something, while \cref{app:1} tells us something else!
\begin{appendices}
\crefalias{section}{appsec}
\section{The first appendix}\label{app:1}
\lipsum[2]
\end{appendices}
\end{document}
\Crefnameand\crefname– OJFord Jun 19 '17 at 13:58\crefalias{subsection}{appsec}, in which case references to subsections in the appendix will have the correct label as well. – Jim Garrison May 30 '18 at 21:35\crefnameis for defining the lowercase form of the reference name and\Crefnameis for defining the uppercase form. It would be better to use\crefname{appsec}{appendix}{appendices}or\Crefname{appsec}{Appendix}{Appendices}in this example to fit with the intent of thecleverefpackage. The package provides the optioncapitalizefor users who wish all references to be capitalized. – RobotRaven Mar 18 '20 at 22:15\Crefnamevariant respects the class option. – Andrew Swann Mar 19 '20 at 08:11