= Solution
Put the anticommuting parameter on the left and write $\delta\Phi=\epsilon s\Phi$. Then $s$ is a <left-acting BRST differential>: it is odd and satisfies the graded product rule $s(UV)=(sU)V+(-1)^{|U|}U(sV)$. The <Faddeev-Popov ghost field> $c$ and <antighost field> $b$ are odd, whereas $A_\mu$ and the <Nakanishi-Lautrup field> $h$ are even. Use $sA_\mu=D_\mu c$, $sc=-[c,c]/2$, $sb=h$, $sh=0$.
A total derivative in the Lagrangian variation must be included in the <Noether current>. For constant $\epsilon$, the ghost covariant derivative has $s(D_\mu c)=0$, by the <Jacobi identity> and the odd statistics of $c$. The Yang-Mills term is invariant, and the remaining variation is
$$
\delta\mathcal L=\epsilon\left[(\partial^\mu h^a)(D_\mu c)^a+h^a\partial^\mu(D_\mu c)^a\right]
=\epsilon\partial_\mu K^\mu,\qquad K^\mu=h^a(D^\mu c)^a.
$$
The term proportional to $\xi$ does not vary because $sh=0$.
To obtain the signs without an ambiguity about fermionic canonical momenta, now allow $\epsilon=\epsilon(x)$. In particular,
$$
\delta(D_\mu c)^a=-\frac12(\partial_\mu\epsilon)f^a{}_{bc}c^bc^c.
$$
The coefficients of $\partial_\mu\epsilon$ in the Yang-Mills, ghost, and multiplier terms are respectively
$$
-\frac1{g^2}F^{\mu\nu a}(D_\nu c)^a,\qquad
h^a(D^\mu c)^a+\frac12(\partial^\mu b^a)f^a{}_{bc}c^bc^c,\qquad
h^a(D^\mu c)^a.
$$
The positive sign of the last ghost expression results from moving $\partial_\mu\epsilon$ past the odd $\partial^\mu b^a$. Subtracting the total-derivative term $K^\mu$ therefore gives the <Yang-Mills BRST Noether current>
$$
\boxed{j_{\mathrm{BRST}}^\mu=-\frac1{g^2}F^{\mu\nu a}(D_\nu c)^a+h^a(D^\mu c)^a+\frac12(\partial^\mu b^a)f^a{}_{bc}c^bc^c.}
$$
Indeed, the full localized variation is $\delta\mathcal L=(\partial_\mu\epsilon)j^\mu+\partial_\mu(\epsilon K^\mu)$. The <Noether theorem> then gives $\partial_\mu j^\mu=0$ on the field equations. With spatial boundary terms vanishing, the corresponding <BRST charge> in four spacetime dimensions is
$$
\boxed{Q_{\mathrm{BRST}}=\int d^3\mathbf x\left[-\frac1{g^2}F^{0\nu a}(D_\nu c)^a+h^a(D^0c)^a+\frac12(\partial^0b^a)f^a{}_{bc}c^bc^c\right].}
$$
It is odd and has ghost number one. Overall generator phases depend on the convention relating this <Noether charge> to quantum commutators; one may use $s\mathcal O=i[Q_{\mathrm{BRST}},\mathcal O\}$. The displayed current fixes the classical Noether normalization for the left-parameter convention.
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