The average size of downward closed family of the subsets of $[n]$ is at most $n/2$?

This is true and indeed, much more can be said: if $\mathcal F$ is a downward closed family of subsets of $[n]$, then $$ \frac1{|\mathcal F|}\, \sum_{F\in\mathcal F} |F| \le \frac12\, \log_2|\mathcal F|; $$ equivalently, if $A\subseteq\{0,1\}^n$ is a downset, then $$ \frac1{|A|}\, \sum_{a\in A} w(a) \le \frac12\, \log_2 |A|, $$ where $w(a)$ is the number of non-zero components of $a$. This is Theorem 3 of the linked paper (see also the abstract).

Here is a pedestrian answer. If $\def\cF{\mathcal F}\cF$ is a downward closed subset of $\mathcal P([n])$, we have

$$\frac1{|\cF|}\sum_{A\in\cF}|A|=\sum_{i\in[n]}\Pr_{A\in\cF}[i\in A].$$

Now, for any $i\in[n]$,

$$\Pr_{A\in\cF}[i\in A]\le\frac12,$$

because the mapping $A\mapsto A\smallsetminus\{i\}$ provides an injection

$$\{A\in\cF:i\in A\}\to\{A\in\cF:i\notin A\}.$$