How to check if the database is in restricted mode

To check what kind of logins that are allowed in your instance, query the LOGINS field of the v$instance view:

select logins from v$instance;

Output here will be either ALLOWED or RESTRICTED

Join with v$database for more information:

select a.INSTANCE_NAME, a.ACTIVE_STATE,a.logins, b.open_mode 
from v$instance a inner join v$database b 
on UPPER(a.instance_name) = b.name;

Which will show the following output if your database is in restricted mode, and in this example, mounted:

INSTANCE_NAME    ACTIVE_ST LOGINS     OPEN_MODE
---------------- --------- ---------- --------------------
mydb01           NORMAL    RESTRICTED MOUNTED

and for normal mode:

INSTANCE_NAME    ACTIVE_ST LOGINS     OPEN_MODE
---------------- --------- ---------- --------------------
mydb01           NORMAL    ALLOWED    READ WRITE