Currently, the only way to identify if a new runner version is available or not is by looking at the runner section in the build log.

Suggestions: 

• Web page with different runner versions along with changelog.
• Optional Auto Update feature for runners.

For example (Windows runner) :
https://product-downloads.atlassian.com/software/bitbucket/pipelines/atlassian-bitbucket-pipelines-runner-latest.zip (should point to the latest runner version)

And version details on specific versions of the runner.