Define a new name format firstinits-last (derived from first-last with \iffirstinits true), and point the labelname format (which is used in citations) to firstinits-last. (This also works if you don't set the firstinits=true package option; with this option enabled, you may simply add \DeclareNameAlias{labelname}{default} to your preamble.)
\documentclass{article}
\usepackage[style=verbose-trad2]{biblatex}
\DeclareNameFormat{firstinits-last}{%
\usebibmacro{name:first-last}{#1}{#4}{#5}{#7}%
\usebibmacro{name:andothers}}
\DeclareNameAlias{labelname}{firstinits-last}
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@misc{A01,
author = {Author, Anton},
year = {2001},
title = {Alpha},
}
@misc{B02,
author = {Buthor, Berta},
year = {2002},
title = {Bravo},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\textheight 160pt% just for the example
\begin{document}
Some text.\autocite{A01}
Some text.\autocite{B02}
Some text.\autocite{A01}
\printbibliography
\end{document}
