Deploy a docker image from a registry with basic auth
cf push currently (version 6.26.0) does not support private docker registries.
This feature is currently being added to the Cloud Controller. It is not even available yet in the latest CF release (v258) but the API is documented in the CC release candidate documentation: http://apidocs.cloudfoundry.org/release-candidate/apps/creating_a_docker_app.html
You can use cf curl to call this API until cf push supports it.
The CLI tracker story to follow is https://www.pivotaltracker.com/story/show/143540893.
Cloud Foundry will support pushing apps from images in private Docker repositories pretty soon, the feature has been released as part of Cloud Controller CC 1.28.0. Expect this feature to be released in CF v260, which should be adopted by providers within the next 4 to 6 weeks.
The Command Line Interface (CLI) has a story in their backlog to support adding username/password as parameter and environment variable to cf push.