The cleanest and simplest way to do this is to use Flexbox! The following will vertically align a Bootstrap 3 modal in the center of the screen and is so much cleaner and simpler than all of the other solutions posted here:
body.modal-open .modal.in {
display: flex !important;
align-items: center;
}
NOTE: While this is the simplest solution, it may not work for everyone due to browser support: http://caniuse.com/#feat=flexbox
It looks like (per usual) IE lags behind. In my case, all the products I develop for myself or for clients are IE10+. (it doesn't make sense business wise to invest development time supporting older versions of IE when it could be used to actually develop the product and get the MVP out faster). This is certainly not a luxury that everyone has.
I have seen larger sites detect whether or not flexbox is supported and apply a class to the body of the page - but that level of front-end engineering is pretty robust, and you'd still need a fallback.
I would encourage people to embrace the future of the web. Flexbox is awesome and you should start using it if you can.
P.S. - This site really helped me grasp flexbox as a whole and apply it to any use case: http://flexboxfroggy.com/
EDIT: In the case of two modals on one page, this should apply to .modal.in