You can either use the psmallmatrix environment from mathtools, or the \medmath command from nccmath (ca 80 % of \displaystyle), or make a group in footnotesize. Just choose:
\documentclass[12pt]{scrartcl}
\usepackage{mathtools, nccmath}
\usepackage{amssymb}
\begin{document}
One standard basis vectors of $\mathbb{R}^2$ is $\begin{pmatrix}1\\0\end{pmatrix}$.
Unfortunately, the vector is about twice as tall as the rest of the line.
How can I reduce its size to about $1.5$-times the size of the line?
One standard basis vectors of $\mathbb{R}^2$ is $\begin{psmallmatrix}1\\0\end{psmallmatrix}$.
Unfortunately, the vector is about twice as tall as the rest of the line.
How can I reduce its size to about $1.5$-times the size of the line?
One standard basis vectors of $\mathbb{R}^2$ is $\medmath{\begin{pmatrix}1\\0\end{pmatrix}}$.
Unfortunately, the vector is about twice as tall as the rest of the line.
How can I reduce its size to about $1.5$-times the size of the line?
?
One standard basis vectors of $\mathbb{R}^2$ is {\scriptsize$\begin{pmatrix}1\\0\end{pmatrix}$}.
Unfortunately, the vector is about twice as tall as the rest of the line.
How can I reduce its size to about $1.5$-times the size of the line?
\end{document}

$\begin{pmatrix}0&1\end{pmatrix}^T$because it is more clearear what are you using. – manooooh Jan 01 '19 at 01:04