\boxed text as subscript

You can use \text from amsmath, but you need to help it as regards to the baseline.

\documentclass{book}
\usepackage{amsmath}
\usepackage{array}
\usepackage{siunitx}

\sisetup{retain-explicit-plus}

\makeatletter
\newcommand{\boxedarray}[2][c]{%
  \begingroup
  \let\current@f@size\f@size
  \setlength\fboxsep{1pt}%
  \text{\fbox{\fix@baseline$\begin{array}[#1]{c}#2\end{array}$}}%
  \endgroup
}
\newcommand{\fixbaseline}{%
  \ifx\f@size\current@f@size\else\ifx\f@size\sf@size\linespread{0.7}\else\linespread{0.5}\fi\fi
  \selectfont
}
\makeatother

\begin{document}

The passenger moving with a velocity of \SI{+11}{m/s}, due in part to the 
walking motion and in part to the train's motion. As
$\boxedarray{ PT \\ AT \\ CT }$ 
the passenger moving with a velocity of \SI{+11}{m/s}, due in part to the 
walking motion and in part to the train's motion. As
\[
\vec{\mathbf{V}}_{\boxedarray{ PT \\ AT \\ CT }}
\]

\end{document}

For units, as already suggested, use siunitx facilities.

I propose a very simple \subboxed command, which ensures the maths will be in \scriptstyle and the value of \boxsep is adjusted. In the maintext, I'd simply use a boxed \Vectorstack from stackengine.

Unrelated to the problem: I used the extensible vector arrows from esvec , which look nicer than \vec with a bold V.

\documentclass{book}
\usepackage{amsmath}
\usepackage{siunitx}
\usepackage[b]{esvect}
\usepackage[usestackEOL]{stackengine}

\newcommand{\subboxed}[1]{{\setlength{\fboxsep}{1pt}\boxed{\everymath{\scriptstyle}#1}}}

\begin{document}

The passenger moving with a velocity of \SI{+11}{\meter/\second}, due in part to the walking
motion and in part to the train's motion. As $\boxed{\Vectorstack{PT \\ AT \\ CT}}$ the passenger moving with a velocity of \SI{+11}{\meter/\second},
due in part to the walking motion and in part to the train's motion. As
\[\vv{\mathbf{V}}_{\subboxed{\substack{PT\\ AT \\ CT}}}
\]

\end{document} 

A method with TABstacks:

\documentclass{book}
\usepackage{amsmath,scalerel,tabstackengine}
\TABstackMath
\TABstackMathstyle{\SavedStyle}

\renewcommand{\boxed}[1]{\ThisStyle{\fboxsep1pt\fbox{%
  \setstackgap{L}{12\LMpt}\tabbedCenterstack{#1}}}}

\begin{document}

the passenger moving with a velocity of +11 m/s, due in part to the walking 
motion and in part to the train's motion. As  $\boxed{PT\\AT\\CT}$ the 
passenger moving with a velocity of +11 m/s, due in part to the walking 
motion and in part to the train's motion. As
\[
\vec{\mathbf{V}}_{\boxed{PT\\AT\\CT}}
\]
\end{document}