If it's the laptop with the bad screen, then I would hook up an external monitor to it first. Then in Mail on the laptop, add an IMAP (Gmail), or Exchange (Microsoft) account. You can then drag and drop all of your mail from your local mailboxes in Mail, to the server mailboxes.
The benefit of using this method is that once you have you mail on an external mail server, then you will be able to abandon Mail altogether and instead begin using webmail from one of those major providers. Webmail provides the ultimate in flexibility and takes away the painful process of managing and migrating local mailboxes when you get new devices. If you really don't like webmail, then you will at least be storing your mail on the server instead on locally in Mail (which, if you read these forums much, is a never ending source of trouble)