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.

Is there any similar characterization of the stronger property of having a $\kappa$-closed dense subset, in terms of what happens in generic extensions?

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.