This is a question that connects to many things in set theory (and they are sometimes called ``strongly almost disjoint families").

First, an old result of Baumgartner (see Section 6 of [1]) shows that you can start with a model of GCH and force the existence of such families for a given $\kappa$ without collapsing cardinals or changing cofinalities. Baumgartner's forcing will result in a model in which $\kappa\leq 2^{\aleph_0}$. He obtains more general results: Assume GCH. Given infinite cardinals $\nu<\kappa<\lambda$ with $\nu$ regular, you can force the existence of a family $\mathcal{A}\subseteq [\kappa]^{\kappa}$ of size $\lambda$ such that any two members of $\mathcal{A}$ have intersection of cardinality less than $\nu$.

On the other hand, ZFC tells us that there are no such $\kappa$ above $\beth_\omega$, the first strong limit singular cardinal (best viewed as the supremum of the sequence $2^{\aleph_0}, 2^{2^{\aleph_0}}, 2^{2^{2^{\aleph_0}}},..$).

If we let $\mu=\beth_\omega$, then by Shelah's Revised GCH Theorem [3], for any $\kappa>\mu$ we can find $\mathcal{P}\subseteq [\kappa]^{<\mu}$ and a regular $\sigma<\mu$ such that

- $|\mathcal{P}|=\kappa$, and
- every $A\in [\kappa]^{<\mu}$ is a union of fewer than $\sigma$ members of $\mathcal{P}$.

(See in particular Conclusion 1.2 (4) from [3].)

Since $\sigma$ is regular, each subset of $\kappa$ of cardinality at least $\sigma$ must have a subset of cardinality at least $\sigma$ lying in $\mathcal{P}$. Thus, given $\mathcal{A}\subseteq[\kappa]^\kappa$ of cardinality $\kappa^+$, we can find $X\subseteq \kappa$ of cardinality $\sigma$ such that
$$|\{A\in\mathcal{A}: X\subseteq A\}|=\kappa^+,$$
which is much stronger than you require.

Now what happens for $\kappa$ between $2^{\aleph_0}$ and $\beth_\omega$? This turns out to be connected to pcf theory, and is very much open.

For example, for a cardinal $\kappa>2^{\aleph_0}$, the following two statements are equivalent:

There is a family $\mathcal{A}\subseteq [\kappa]^{\aleph_1}$ of cardinality $>\kappa$ with pairwise finite intersection.

There is a sequence $\langle A_\alpha:\alpha<\omega_1\rangle$ such that each $A_\alpha$ is a finite collection of regular cardinals from the interval $(2^{\aleph_0},\kappa]$, and such that for every infinite $X\subseteq \omega_1$, ${\rm maxpcf}(\bigcup_{\alpha\in X}A_\alpha)>\kappa$.

The above is a theorem of Shelah (see Section 6 of [2]). It is unknown if this is consistent, but the existence of such a family entails some drastic failures of the singular cardinals hypothesis.

[1] *Baumgartner, James E.*, **Almost-disjoint sets, the dense set problem and the partition calculus**, Ann. Math. Logic 9, 401-439 (1976). ZBL0339.04003.

[2] *Shelah, Saharon*, **More on cardinal arithmetic**, Arch. Math. Logic 32, No. 6, 399-428 (1993). ZBL0799.03052.

[3] *Shelah, Saharon*, **The generalized continuum hypothesis revisited**, Isr. J. Math. 116, 285-321 (2000). ZBL0955.03054.