Starting with the September 27 Open Beta, we are using new Wizards Accounts for the game. If you had an account on Closed Beta, the first time you log into the MTG Arena client, you'll be prompted to update your account for the new system.
If you don't remember your password from Closed Beta, you can reset that password at https://mtgarenapasswordreset.wizards.com.
Once you've logged into the client for the first time, you should use https://myaccounts.wizards.com/forgot to reset your password going forward (or just the Need Help -> Forgot your Password option in the client).
In short, use whatever email address you used in Closed Beta to log into the client, and you will be transitioned to the system for Open Beta!