How to change the default border color of fbox?

You can try this. It will also set the background to white, normally this should be not a problem.

\documentclass{article}

\usepackage{xcolor}

\renewcommand\fbox{\fcolorbox{red}{white}}
\begin{document}

\fbox{text}

\end{document}

Without setting the background color, one can save the current text color and apply it when the text of the \fbox is typeset, changing color for typesetting the rules.

\documentclass{article}
\usepackage{xcolor}
\usepackage{etoolbox}

\makeatletter
\let\cfbox\fbox
\let\c@frameb@x\@frameb@x
\pretocmd{\cfbox}
  {\leavevmode\begingroup\colorlet{currentcolor}{.}\color{red}}
  {}{}
\patchcmd\cfbox{\@frameb@x}{\c@frameb@x}{}{}
\patchcmd{\c@frameb@x}
  {\box\@tempboxa}
  {\color{currentcolor}\box\@tempboxa}
  {}{}
\apptocmd{\c@frameb@x}{\endgroup}{}{}
\makeatother

\begin{document}

text \cfbox{text} text

\fcolorbox{green}{blue}{text \cfbox{text} text}

\end{document}

I'd prefer a different command to redefining \fbox. If you instead want to use \fbox, just remove the \let\cfbox\fbox line and change \cfbox into \fbox in the remaining places.