= Solution
Let $f$ be <probability density> per unit solid angle, with $\int_{S^2}f\,d\Omega=1$ and isotropic initial condition $f(\mathbf p,0)=1/(4\pi)$. In still fluid, <bottom-heavy spherical-cell orientation dynamics> gives $\dot{\mathbf p}=B^{-1}[\mathbf k-(\mathbf k\cdot\mathbf p)\mathbf p]$. Adding isotropic rotational <diffusion> gives the <gyrotactic orientation Fokker-Planck equation>
$$
\partial_tf+\nabla_p\cdot(f\dot{\mathbf p})=D_r\Delta_{S^2}f.
$$
The derivatives are tangential to the unit sphere. Rotational symmetry about $\mathbf k$ is preserved by both the equation and the initial condition, so $f$ depends only on $\theta,t$. Since $\dot\theta=-\sin\theta/B$, the polar probability flux is $j_\theta=-\sin\theta f/B-D_rf_\theta$. Expanding $\partial_tf+\sin^{-1}\theta\,\partial_\theta(\sin\theta j_\theta)=0$ yields
$$
\boxed{Bf_t-\sin\theta f_\theta-2\cos\theta f
=\frac{BD_r}{\sin\theta}\partial_\theta(\sin\theta f_\theta).}
$$
Regular solutions have no probability flux through the poles. The mean swimming <velocity> is $\mathbf V_c=V_s\langle\mathbf p\rangle$, with only a vertical component by symmetry.
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