Should Auto_Close be ON for any of your databases? In short, no. Having the Auto_Close option ON = increased overhead. Or, per MSFT, "...this option can cause performance degradation on frequently accessed databases because of the increased overhead of opening and closing the database after each connection."
In my book that means turn it off. Here's a quick query to show you which databases have Auto_Close ON:
-- is auto_close enabled
SELECT
name [Database],
CASE WHEN is_auto_close_on =
0 THEN 'No'
WHEN
is_auto_close_on = 1 THEN 'Yes' END AutoCloseEnabled
FROM sys.databases
And this is what you use to set it OFF:
-- set auto_close off
ALTER DATABASE ReportServer
SET AUTO_CLOSE OFF
Take a look at both of these for more details:
Set the AUTO_CLOSE Database Option to OFF