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Partition the enumeration of $B$ into the pairs $\{b_{2j-1},b_{2j}\}$, for $j\ge 1$. An infinite binary sequence $x$ determines a bijection $h_x:B\to B$: leave the $j$th pair fixed when $x_j=0$, and interchange its two elements when $x_j=1$. Each element stays in its own pair, so this function is a bijection, indeed its own inverse.

Different binary sequences give different bijections, since a changed bit changes the action on the corresponding pair. We have therefore embedded an uncountable set into $S_3$. Hence $S_3$ is uncountable, proving uncountability of permutations of a countably infinite set.