Running Selenium test suite in a Continuous Environment has many advantages. The test suite can be automatically triggered upon a code commit to Git, SVN etc. You can also run the test suite periodically based on the desired time of the day. Test run results can then be emailed, sent via SMS(only pass/fail status), or send to Hipchat, or Slack client.
The best way to see the selenium test run if a test fails is via video capture. Saucelabs, and Browserstack both have video playback options. You can simply take the failed session URL, and past it to Jenkins next to the failed test. This simplifies debugging a failed test. You can just click on the link, and watch where the test failed. Here’s an example of how the failed test URLs are displayed on Jenkins.

You can also get the Jenkins test pass/fail status by calling the Jenkins API directly. Here’s an example of how you can get the result using REST API.
require 'json'
require 'rest-client'
url = "https://username:Password@#{jenkins_url}/lastCompletedBuild/api/json"
response = JSON.parse(RestClient.get(url))
result = response["result"]