A closely related problem, considering the $(n-1)$-dimensional surface area instead of the diameter of the body, has been solved by Keith Ball, *Volume ratios and a reverse isoperimetric inequality*. J. London Math. Soc. (2) 44 (1991), no. 2, 351–359 [MR1136445]. The extreme case, as expected, is the $n$-dimensional simplex. Ball's proof may also work for the diameter. The article is available on the arXiv, see http://arxiv.org/abs/math/9201205 . The 2-dimensional case was solved long before, with a very short, elementary proof.

A quick estimate of the diameter can be obtained by taking the minimum volume ellipsoid containing the body $K$ and using the known (best possible, by the way) estimate of the ellipsoid's volume. An affine transformation that turns the ellipsoid into a ball yields a bound on the diameter - perhaps not the best possible, but a fairly good one.