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 further
splitting conditions.  There is an extensive literature on divisibility and indivisibility of class numbers and Kimura's paper has many other relevant references.  

**Update**  There is also an [erratum][2] to Kimura's paper -- it seems that he needs the existence of one such field to get a lower bound for the number of such fields; for large $p$ this is guaranteed by work of Horie.  In the meantime a paper of [Wiles][3] has addressed the general version of this question, and there is a quantification of Wiles's result due to [Beckwith][4].

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

[2]: https://www.impan.pl/pl/wydawnictwa/czasopisma-i-serie-wydawnicze/acta-arithmetica/all/114/4/82517/correction-to-8220-a-note-on-the-existence-of-certain-infinite-families-of-imaginary-quadratic-fields-8221-acta-arith-110-2003-37-8211-43

[3]: http://onlinelibrary.wiley.com/doi/10.1112/jlms/jdv031/abstract

[4]: https://arxiv.org/pdf/1612.04443.pdf