= Solution
Taking the <Fourier transform of a derivative> gives
$$
iξ\mathbin\cdot\widehat B=\widehat f,\qquad ξ\mathbin\times\widehat B=0.
$$
Thus, away from $ξ=0$,
$$
\widehat B_j(ξ)=-iξ_j|ξ|^{-2}\widehat f(ξ).
$$
This formula determines the $L^2$ solution uniquely; a distribution supported only at $ξ=0$ cannot belong to $L^2$ unless it is zero.
For $|ξ|\leq1$, the <Fourier transform> estimate $|\widehat f(ξ)|\leq C\|f\|_1$ and local integrability of $|ξ|^{-2}$ in three dimensions control the squared norm. For $|ξ|>1$,
$$
(1+|ξ|^2)^{s+1}|\widehat B_j|^2\leq C(1+|ξ|^2)^s|\widehat f|^2.
$$
The <Plancherel theorem> therefore yields $\|B_j\|_{H^{s+1}}\leq C(\|f\|_1+\|f\|_{H^s})$. Finally the <Sobolev embedding theorem> in three dimensions gives $H^{s+1}\subset C^1$ when $s+1>1+3/2$, namely when $s>3/2$.
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