Answer: D

Solution:

Let the outer square have area $1.$ The nested squares have areas $1, k^2, k^4, \ldots,$ so the ring between the $n$th and $(n+1)$th squares has area $k^{2n}(1-k^2).$

The shaded rings are the alternate ones $n = 0, 2, 4, \ldots,$ with total area $$\begin{aligned} \sum_{j=0}^{\infty} k^{4j}(1-k^2) &= \frac{1-k^2}{1-k^4} \\ &= \frac{1}{1+k^2}. \end{aligned}$$

Setting $\dfrac{1}{1+k^2} = \dfrac{16}{25}$ gives $1 + k^2 = \dfrac{25}{16},$ so $k^2 = \dfrac{9}{16}$ and $k = \dfrac{3}{4}.$

Thus, the correct answer is D.