 Solution

This is the Rasiowa–Sikorski lemma. Put $p_0=p$ and recursively chooseusing density. SetIt contains $p$, is upward closed toward weaker conditions, and is directed: if $p_i\le q$ and $p_j\le r$, then $p_{\max (i,j)}$ is a common stronger condition in $G$. It meets each $D_n$ through $p_{n+1}$. Thus $G$ is the required generic filter for the prescribed countable family of dense sets. For a countable transitive model, externally enumerate its dense sets to obtain a generic filter over the model by this same argument.