How to style blockquote?

Assuming you use the setspace package and the instruction \onehalfspacing to modify the line spacing of the document, you may achieve your objective by including the following instructions in the preamble:

\usepackage{etoolbox}
\AtBeginEnvironment{quote}{\singlespacing\small}

In the document classes I'm familiar with, if the main document font size is 12pt, the instruction \small switches to 11pt as the font size.

A full MWE (minimum working example):

\documentclass[12pt]{article}
\usepackage{lipsum}   % for filler text
\usepackage{setspace} % for \onehalfspacing and \singlespacing macros
\onehalfspacing 

\usepackage{etoolbox}
\AtBeginEnvironment{quote}{\singlespacing\small}

\begin{document}
\lipsum[2]  % filler text
\begin{quote}
\lipsum[2]
\end{quote}
\lipsum[2]
\end{document}

Since this is the first result I got when I Googled "latex block quote", I thought I'd mention the csquotes package I found via this Overleaf article. Seems to do about the same thing, just not the font resizing the OP asked for.

\documentclass[12pt]{book}
\usepackage[english]{babel}
\usepackage{csquotes}
\usepackage{lipsum}

\begin{document}

\lipsum[1]

\begin{displayquote}
\lipsum[2]
\end{displayquote}

\lipsum[3]

\end{document} 

... gets you this....

You can do that with the etoolbox package:

 \documentclass[12pt]{book}
\usepackage{setspace}
\usepackage{lipsum}
\usepackage{etoolbox}
\AtBeginEnvironment{quote}{\singlespace\vspace{-\topsep}\small}
\AtEndEnvironment{quote}{\vspace{-\topsep}\endsinglespace}

\onehalfspacing

\begin{document}

\lipsum[1]
\begin{quote}
\lipsum[2]
\end{quote}

\lipsum[3]

\end{document}