\blockquote is actually a dynamic command that switches between typesetting its argument inline (if it is shorter than a threshold set with the package options threshold and thresholdtype) or in display mode (i.e. as a real block quotation separated by line/paragraph breaks). These two modes can be modified independently.
If you always want to typeset block quotes regardless of the length, maybe \begin{displayquote}...\end{displayquote} is more to the point. The behaviour of that environment can be controlled with \SetBlockEnvironment amongst others.
Specifically for the job of making block quotes smaller csquotes.cfg suggest to define a new environment based on quote that can then be used for \blockquote with \SetBlockEnvironment.
\documentclass[french]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{csquotes}
\usepackage{babel}
\usepackage{relsize}
\newenvironment*{smallquote}
{\quote\smaller}
{\endquote}
\SetBlockEnvironment{smallquote}
\usepackage{lipsum}
\begin{document}
\lipsum[1]
\blockquote[somebody]{Lorem ipsum. A short inline quote.}
\lipsum[2]
\blockquote[somebody]{\lipsum[3]}
\lipsum[4]
\end{document}

Alternatively, you can use the hooks provided by csquotes.
The inline quotation can be modified by redefining the hook \mktextquote, the block quotation by changing \mkblockquote.
It would look odd to have inline quotations in a different font size, so I'm assuming you only want to change the block quotation. The default definition for \mkblockquote is
\newcommand{\mkblockquote}[4]{#1#2#4#3}
we just add a \smaller (from \usepackage{relsize} as suggested in Tiuri's answer) and then use \renewcommand.
\documentclass[french]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{csquotes}
\usepackage{babel}
\usepackage{relsize}
\renewcommand{\mkblockquote}[4]{\smaller#1#2#4#3}
\usepackage{lipsum} % for filler text
\begin{document}
\lipsum[1]
\blockquote[somebody]{Lorem ipsum. A short inline quote.}
\lipsum[2]
\blockquote[somebody]{\lipsum[3]}
\lipsum[4]
\end{document}
The output in this example is as shown above.
\SetBlockEnvironment also affects \begin{displayquote}...\end{displayquote}, but for finer control those environments can be customised with \mkbegdispquote and \mkenddispquote, whose defaults are
\newcommand{\mkbegdispquote}[2]{}
\newcommand{\mkenddispquote}[2]{#1#2}
So we could also use
\documentclass[french]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{csquotes}
\usepackage{babel}
\usepackage{relsize}
\renewcommand{\mkbegdispquote}[2]{\smaller}
\usepackage{lipsum} % for filler text
\begin{document}
\lipsum[1]
\begin{displayquote}[somebody]
Lorem ipsum. A short inline quote.
\end{displayquote}
\lipsum[2]
\begin{displayquote}[somebody]
\lipsum[3]
\end{displayquote}
\lipsum[4]
\end{document}
Bla bla bla.,« Bla bla bla. », or« Bla bla bla. » (somebody)? – ebosi May 16 '17 at 15:39