Actually, "having an $\sigma$-closed dense subset" is not a very good forcing invariant. In his paper On the existence of a $\sigma$-closed dense subset [Comment. Math. Univ. Carolin. 51 (2010), 513-517; MR2741884], Jindra Zapletal shows that it is relatively consistent with ZFC that there is a partial order $(P \cup Q,{\leq})$ in which $P$ and $Q$ are both dense, $(P,{\leq})$ is $\sigma$-closed but $(Q,{\leq})$ has no $\sigma$-closed dense subset. It is an open problem whether such an example provably exists in ZFC.

This example shows that a nice characterization is unlikely in general. However, Matt Foreman [J. Symbolic Logic 48 (1983), 714-723; MR0716633] and Petr Vojtáš [Comment. Math. Univ. Carolin. 24 (1983), no. 2, 349–369; MR0711272] have found some useful partial results.

Actually, "having a $\kappa$-closed dense subset" is not (necessarily) a forcing invariant. In his paper On the existence of a $\sigma$-closed dense subset [Comment. Math. Univ. Carolin. 51 (2010), 513-517; MR2741884], Jindra Zapletal shows that it is relatively consistent with ZFC that there is a partial order $(P \cup Q,{\leq})$ in which $P$ and $Q$ are both dense, $(P,{\leq})$ is $\omega_1$-closed but $(Q,{\leq})$ has no $\omega_1$-closed dense subset. It is an open problem whether such an example provably exists in ZFC.

This example shows that a nice characterization is unlikely in general. However, Matt Foreman [J. Symbolic Logic 48 (1983), 714-723; MR0716633] and Petr Vojtáš [Comment. Math. Univ. Carolin. 24 (1983), no. 2, 349–369; MR0711272] have found some useful partial results.