find ./src_dir -type f -not -path '*/.*' will find all files in src_dir excluding the ones where the path contain a . at the beginning of a file or folder.
xargs cp --parents -t ./dest_dir will copy the files found to dest_dir, recreating the folder hierarchy thanks to the --parents argument.
Note: This will not copy empty folders. And will effectively exclude all hidden files and folders from being copied.