= Solution
Use <mass dimensions> with $[x]=-1$, $[\partial_\mu]=1$ and dimensionless <action>. The <Lagrangian density> in <Euclidean space> has dimension $D$, and the <kinetic term> fixes $2+2[\phi_0]=D$. Thus
$$
[\phi_0]=\frac{D-2}{2},\qquad
\boxed{[\lambda_{2k}]=D-k(D-2)=2k-(k-1)D\equiv d_k.}
$$
A dimensionless <coupling constant (physics)> is therefore
$$
\boxed{g_{2k}(\mu)=\mu^{-d_k}\lambda_{2k},\qquad
\lambda_{2k}=\mu^{d_k}g_{2k}.}
$$
Here $\mu$ is a positive <regularization> scale with <mass dimension> one. The interaction is marginal at its <upper critical dimension> $D_c=2k/(k-1)$, relevant below it and irrelevant above it by this engineering-dimension test. In particular, $d_2=4-D$, while $d_k=4-2k+2(k-1)\epsilon$ near $D=4-2\epsilon$. At this stage $g_{2k}$ is the dimensionless version of the displayed bare coupling; after introducing <counterterms> one may use the same scale factor for the <renormalized coupling>.
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