If the radius of the base of a right circular cylinder is halved,

Let the radius and height of the original cylinder be R and h, respectively.

Now, the radius of the new cylinder $=\frac{R}{2}$

Then, the ratio of the volume of the cylinder thus obtained to the volume of original cylinder is given by

$\pi\left(\frac{R}{2}\right)^{2} h: \pi R^{2} h$

$=\frac{1}{4}: 1$

$=1: 4$

Hence, the correct answer is option C.