jmc

The four points $(0,0),$ $(0,2),$ $(3,0),$ and $(3,2)$ form a rectangle, and the horizontal line through $(-\frac{3}{2},1)$ bisects the rectangle. So, visually, we hope that the center of the ellipse coincides with the center of the rectangle, which has coordinates $\left(\frac{3}{2},1\right),$ and that its major axis should pass through the point $(-\frac{3}{2},1).$

In this case, the semimajor axis has length $\frac{3}{2}-(-\frac{3}{2})=3.$ Then, its equation must take the form $$\frac{(x-\frac{3}{2}{)}^2}{3^2}+\frac{(y-1{)}^2}{b^2}=1$$where $b$ is the length of the semiminor axis. Since $(0,0)$ lies on the ellipse, setting $x=y=0,$ we have $$\frac{{\left(\frac{3}{2}\right)}^2}{3^2}+\frac{1}{b^2}=1,$$or $\frac{1}{4}+\frac{1}{b^2}=1.$ Solving for $b$ gives $b=\frac{2\sqrt{3}}{3},$ so the length of the minor axis is $2b=\boxed{\frac{4\sqrt{3}}{3}}.$