The existing UI is not designed to be responsive, and is not optimized to work on mobile form-factors as such.
However, it is built on SocialMiner chat REST APIs which are published and available. 3rd-party developers can definitely use these REST APIs and build their own mobile-based chat UIs for their customers.
See Cisco SocialMiner Developer Guide Release 10.6(1) (Chapter 8) for details on these APIs.
Further, we have hosted a working sample customer chat application (implemented using javascript) on developer.cisco.com which demonstrates the use of SocialMiner chat REST APIs.
See the README within the sample.