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Unconditional? Certainly not, if I understood your question correctly. E.g., suppose that $\mu$ is the standard Gaussian measure on $\mathbb R^d$ and $B=\{0\}$. Then for any $\delta\in(0,1)$, by the law of large numbers, $$1-\mu(B^\epsilon)=P\Big(\frac1d\sum_1^d Z_i^2>\frac{\epsilon^2}d\Big)\to1$$ for $d\to\infty$ and $\epsilon\in(0,(1-\delta)\sqrt d\,]$, where $Z_1,Z_2,\dots$ are iid $N(0,1)$. So, asymptotically there is no decrease at all in $\epsilon\le(1-\delta)\sqrt d$.