How can I deploy a node.js app to a private server after a successful Travis build/test?

According to travis-ci documentation

You can easily deploy to your own server the way you would deploy from your local machine by adding a custom after_success step.

You may choose the Script provider instead, as it provides easier flexibility with conditional deployment.

FTP

env:
  global:
    - "FTP_USER=user"
    - "FTP_PASSWORD=password"
after_success:
    "curl --ftp-create-dirs -T uploadfilename -u $FTP_USER:$FTP_PASSWORD ftp://sitename.com/directory/myfile"

The env variables FTP_USER and FTP_PASSWORD can also be encrypted.

See curl(1) for more details on how to use cURL as an FTP client.

or Git

after_success:
  - eval "$(ssh-agent -s)" #start the ssh agent
  - chmod 600 .travis/deploy_key.pem # this key should have push access
  - ssh-add .travis/deploy_key.pem
  - git remote add deploy DEPLOY_REPO_URI_GOES_HERE
  - git push deploy

See “How can I encrypt files that include sensitive data?” if you don’t want to commit the private key unencrypted to your repository.