From what you said, I'm guessing that this is a POP3 account, not IMAP and that you have your computer's email client incoming server settings to leave mail on the server. So, your phone downloads it as new mail.
You can change your computer to delete mail on the server, but then mail downloaded on the computer will not appear on the phone. You can also do the same the opposite way - set the phone to delete read mail on the server.
It's up to you which behavior you prefer, leave read mail or delete it. If you can change the account to IMAP service you can have much better control over mail handling.
If the account is already IMAP service disregard the babbling above. The default action for IMAP is read mail stays on the server, but then appears as read mail on other devices, not new mail as you stated.
