Make sure your application/blocks folder has read and write permissions (i.e. 0777 rights). You can either do this in your admin/cPanel or via your FTP application (FileZilla or something else). When using this package on your localhost (WAMP/XAMPP/LAMP) it should usually work instantly!
As a default icon gets included with each every block you create (without uploading your own icon), give read/write permissions to the packages/block_designer/img folder aswell to be able to copy and move this file to the newly created block type.
From version 2.9 of Block Designer and upwards, PHP 5.4 is the minimum. So be sure your hosting isn't using any old 5.3.x or lower version(s)!