Looking into jurabib.sty, I stumbled upon a suspicious code block with lots of hardcoded comma signs. Replacing them with space commands indeed did the trick. That doesn't mean I know what all those macros do, and it also doesn't mean you shouldn't consider switching to biblatex. ;-)
\documentclass{article}
\usepackage{jurabib}
\renewcommand\jbRevedFirstOnlyLast{\biblnfmt{\jbLast}}%
\renewcommand\jbRevedFirstNoVonNoJr{\biblnfmt{\jbLast\ }\bibfnfmt{\jbCheckedFirst}}%
\renewcommand\jbRevedFirstNoVonJr{\biblnfmt{\jbLast\ \jbJunior\ }\bibfnfmt{\jbCheckedFirst}}%
\renewcommand\jbRevedFirstVonNoJr{\biblnfmt{\jbLast\ }\bibfnfmt{\jbCheckedFirst\space\jbVon}}%
\renewcommand\jbRevedFirstVonJr{\biblnfmt{\jbLast\ \jbJunior\ }\bibfnfmt{\jbCheckedFirst\space\jbVon}}%
\renewcommand\jbRevedNotFirstOnlyLast{\biblnfmt{\jbLast}}%
\renewcommand\jbRevedNotFirstNoVonNoJr{\bibfnfmt{\jbCheckedFirst\space}\biblnfmt{\jbLast}}%
\renewcommand\jbRevedNotFirstNoVonJr{\bibfnfmt{\jbCheckedFirst\space}\biblnfmt{\jbLast\ \jbJunior}}%
\renewcommand\jbRevedNotFirstVonNoJr{\bibfnfmt{\jbCheckedFirst\space\jbVon~}\biblnfmt{\jbLast}}%
\renewcommand\jbRevedNotFirstVonJr{\bibfnfmt{\jbCheckedFirst\space\jbVon~}\biblnfmt{\jbLast\ \jbJunior}}%
\renewcommand\jbNotRevedOnlyLast{\biblnfmt{\jbLast}}%
\renewcommand\jbNotRevedNoVonNoJr{\biblnfmt{\jbLast\ }\bibfnfmt{\jbCheckedFirst}}% <-- THE IMPORTANT CHANGE
\renewcommand\jbNotRevedNoVonJr{\biblnfmt{\jbLast\ \jbJunior\ }\bibfnfmt{\jbCheckedFirst}}%
\renewcommand\jbNotRevedVonNoJr{\biblnfmt{\jbLast\ }\bibfnfmt{\jbCheckedFirst\space\jbVon}}%
\renewcommand\jbNotRevedVonJr{\biblnfmt{\jbLast\ \jbJunior\ }\bibfnfmt{\jbCheckedFirst\space\jbVon}}%
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@misc{A01,
author = {Author, A.},
year = {2001},
title = {Alpha},
}
\end{filecontents}
\begin{document}
\nocite{*}
\bibliographystyle{jurabib}
\bibliography{\jobname}
\end{document}
