 Solution

Translation by $e=(0,1,0,\dots ,0)$ preserves $H$, so $(\omega (x+e):x\in H)$ has the same independent Bernoulli-$p$ law as $\omega$. This shift is ergodic, and the number $N$ of infinite clusters is shift-invariant. Thus $N=k_0$ almost surely for some deterministic $k_0\in \mathbb{N}\cup \{\infty \}$.

Suppose $2\le k_0<\infty$. With positive probability a finite box meets all $k_0$ infinite clusters. By the finite-energy property of Bernoulli percolation, forcing finitely many sites in the box open has positive conditional probability and joins those clusters without affecting infinity outside the box. The resulting configuration has fewer than $k_0$ infinite clusters on an event of positive probability, contradicting the almost-sure constancy. Hence $k_0\in \{0,1,\infty \}$.