Characterization of $\kappa$-closed forcings For the notion of distributivity of forcings, we have equivalent defintions, one combintorial, the other in terms of what the generic extensions look like.  For a partial order $\mathbb{P}$, the following are equivalent:
1) Forcing with $\mathbb{P}$ adds no new functions from ordinals to ordinals with domain $\kappa$.
2) The intersection of $\kappa$ many dense open subsets of $\mathbb{P}$ is dense.
$\kappa$-closure of partial orders is definitely not a forcing invariant, as witnessed just by the fact that an atomless complete boolean algebra is never countably closed.  However, we can ask is there a characterization partial orders $\mathbb{P}$ in terms of what happens in their generic extensions that is equivalent to the following: The boolean completion of $\mathbb{P}$ has a $\kappa$-closed dense subset?
EDIT: Changed the question in light of the example mentioned by Dorais.
 A: 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.
A: Jensen showed the boolean completion of $\mathbb P$ has a countably closed dense set if and only if $\mathbb P$ satisfies the following property:  For all regular $\theta > | \mathbb P|$, there is a club of countable $M \prec H_\theta$ such that every $(M,\mathbb P)$-generic filter $g$ has a lower bound in $\mathbb P$.
See: MR2840749  Jensen, Ronald . Subcomplete forcing and ℒ-forcing.
E-recursion, forcing and C∗-algebras,
83--182, Lect. Notes Ser. Inst. Math. Sci. Natl. Univ. Singap., 27, World Sci. Publ., Hackensack, NJ,  2014.
