Assuming you are using Drupal 7.
Use the Views module and create a table view of node comments.
Add fields: Node ID (nid), Comment ID (cid), Comment Title, Comment Body, Commment Author, Created Date.
For export, use Views Data Export and export the CSV file.
For import, use the Feeds module + Feeds Comment Processor and import your CSV.
Checkout this Feeds Video Tutorial (this is for nodes, not comments, but it will give you an idea on how feeds with the CSV import works).
Exporting MySQL tables approach:
Never done it like this, but if I were you, I would look for all the tables that have the name comment, export those and import.
Note sure if it will work as simply as that, but worth a try.