ReadCube Papers Mobile retains a significant amount of its functionality when used offline - you can still access your library, annotate, and read articles. You will not be able to sync your library or search on PubMed/Google Scholar when offline, though.
To download an article for offline viewing either click the Download From Cloud button from your library or viewing the description of the article.
You can also swipe on the left side of the application to manually select multiple entries within your library which you can select and decide to download.
We are adding the feature to download your whole library in a future update.