A more general version of this statement was shown by [Kimura][1] in Acta Arith. (2003).  His corollary gives $\gg \sqrt{X}/\log X$ such fields ${\Bbb Q}(\sqrt{-d})$ with $d\le X$, and also allows you to add additional 
splitting conditions.  There is an extensive literature on divisibility and indivisibility of class numbers and Kimura's paper has many other relevant references.

[1]: https://eudml.org/doc/278153