You can easily see if a mailbox database is mounted, when you open your Exchange Management Console, select Server Configuration, and click on your Mailbox server. But, by using the power of the Exchange Management Shell, you just need to execute one line to get a list of all mailbox databases, and if they are mounted or not. The following line will present you with a list of mailbox databases, sorted by servername first, and then by storage group name, and if the mailbox database is mounted or not.
Get-MailboxDatabase -Status | Sort-Object Server | Format-Table Server,StorageGroup,Name,Mounted -wrap