SharePoint does not provide an option to export crawl logs from the Search crawl.
The following set of queries can be executed in the database to export the list of errors from the crawl logs.
select * from MSSCrawlErrorList with (nolock)
This query will return the different types of error that can be generated by the search crawl. Keep a note of the ErrorID for the error that you want to export
Example: ErrorID 692 is for “The crawler could not communicate with the server. Check that the server is available and that the firewall access is configured correctly.”
select * from MSSCrawlHostList with (nolock)
This query will return the web applications that are configured to be crawled.
Example: HostID 2 is for the web application for which you want to export the crawl logs.
Once you get the ErrorID and HostID from the above two queries, execute the following query to export the errors from the crawl log.
select * from MSSCrawlURLLog with (nolock) where ErrorID = 692 and HostID = 2
You can export the results to a text file or in a csv for further analysis.