To determine your website banned by Google, you should check your log files or check with Google analytics report.
If your staff has advanced technical skill, then they can easily find the problems by reviewing log files.
If your staff doesn’t have the advanced technical skill, it is better that you can check the web analytics reports.
If you find a significant drop in Google crawling, then it means that your site has been banned or Google has difficulty crawling your site due to technical reasons.
You can also find with the index count method.
If your site is included in the Google index, it has not been banned. However, if your site has an index count of zero, it is a strong indication that your site might be banned.
Similarly you can check with link count method.
If there are links to your site in the Google index, your site has not been banned. If your site has a link count of zero, it is a strong indication that your site might be banned.
If everything is fine, then find some problems that are in robots.txt file, URL structure, server redirection, site navigation, cross-linking or any password protection issue.
You can also change the web content if your website is banned by Google panda or can request Google to remove the poor backlink using the Google disavow tool if banned by the Google penguin update.