= Solution
This is the <Rasiowa–Sikorski lemma>. Put $p_0=p$ and recursively choose
$$
p_{n+1}\leq p_n,\qquad p_{n+1}\in D_n,
$$
using density. Set
$$
\boxed{G=\{q\in P:\exists n\ p_n\leq q\}.}
$$
It contains $p$, is upward closed toward weaker conditions, and is directed: if $p_i\leq q$ and $p_j\leq r$, 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.
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