There is no way to do this in Boost itself as there is not a block region on the left.
You would need to create a child theme with a new block region on the left and while block regions can be added as required (check out handlebar as one example of a boost child theme with additional block regions - there are others) adding one to the left side of the screen would involve planning what to do with the admin/navigation region - or maybe adding the new block region into that area, where the Add a Block block defaults to.
You could probably add a region that pushes across, as the main region does when the flat navigation region is displayed, or possibly arrange it so it stays where it is and hides behind the pull out region.
Whatever you decide to try, it would need to be a child theme rather than Boost itself, or you would risk losing any changes next time you upgrade Moodle.