Choice A is correct. The standard form of an equation of a circle in the xy-plane is ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$, where the coordinates of the center of the circle are $\left(h,k\right)$ and the length of the radius of the circle is $r$. The equation of circle A, $x^2+{\left(y-2\right)}^2=36$, can be rewritten as ${\left(x-0\right)}^2+{\left(y-2\right)}^2=6^2$. Therefore, the center of circle A is at $\left(0,2\right)$ and the length of the radius of circle A is $6$. If circle A is shifted down $4$ units, the y-coordinate of its center will decrease by $4$; the radius of the circle and the x-coordinate of its center will not change. Therefore, the center of circle B is at $\left(0,2-4\right)$, or $\left(0,-2\right)$, and its radius is $6$. Substituting $0$ for $h$, $-2$ for $k$, and $6$ for $r$ in the equation ${\left(x-h\right)}^2+{\left(y-k\right)}^2=r^2$ yields ${\left(x-0\right)}^2+{\left(y-\left(-2\right)\right)}^2={\left(6\right)}^2$, or $x^2+{\left(y+2\right)}^2=36$. Therefore, the equation $x^2+{\left(y+2\right)}^2=36$ represents circle B.

Choice B is incorrect. This equation represents a circle obtained by shifting circle A up, rather than down, $4$ units.

Choice C is incorrect. This equation represents a circle obtained by shifting circle A right, rather than down, $4$ units.

Choice D is incorrect. This equation represents a circle obtained by shifting circle A left, rather than down, $4$ units.