= Solution
Fix the finite set $S=\{x_1,\ldots,x_n\}\subset B$. If an infinite half-space cluster met $B$ exactly in $S$ with positive probability, take independent reflected occurrences in $H$ and $H'$ and force the finitely many intervening sites adjacent to $S$ closed. The <finite-energy property of Bernoulli percolation> gives this combined event positive probability, while it creates two distinct whole-space infinite clusters, contradicting uniqueness. Thus the probability is zero for every finite $S$. Since $B$ has only countably many finite subsets, almost surely every infinite cluster that meets $B$ meets it infinitely often.
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