How to change the fontsize of a csquotes' quotation?

One idea of the csquotes package is to dynamically decide whether a quote is so long that it is typed as a blockquote, or whether it is short enough to be typed inline. I assume that you don't want to change the font size of inline quotes, but rather of block quotes.

To do that, you need to know that the \blockquote just calls the quote environment, so we just need to change the formatting settings for this environment. This example shows how to do that using the packages etoolbox and relsize:

\documentclass[french]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{csquotes}
\usepackage{babel}

\usepackage{relsize,etoolbox}
\AtBeginEnvironment{quote}{\smaller}

\begin{document}
Bla bla bla.

\blockquote[somebody]{Bla bla bla.\\}

Bla bla bla.
\end{document}

(Note that I added the linebreak \\ only to trigger that csquotes typesets the quote as a blockquote.)

I use the following renewed environment with csquotes. Copy it to your preamble.

\renewenvironment{quote}{\list{}{\rightmargin0pt\leftmargin0pt}\item\relax
\small\singlespacing}{\endlist}
\SetBlockEnvironment{quote}

\rightmargin0pt\leftmargin0pt declares the margins of the quotation (0pt each indicates that is textwidth). What you are looking for is the command \small which sets the quotation one step smaller than the normal text.