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Choice B is correct. $10\%$ of a quantity means $\frac{10}{100}$ times the quantity. Therefore, $10\%$ of $370$ can be represented as $\frac{10}{100}\left(370\right)$, which is equivalent to $0.10\left(370\right)$, or $37$. Therefore, $10\%$ of $370$ is $37$.

Choice A is incorrect. This is $10\%$ of $270$, not $10\%$ of $370$.

Choice C is incorrect. This is $90\%$ of $370$, not $10\%$ of $370$.

Choice D is incorrect. This is $370-10$, not $10\%$ of $370$.