How do I check that the database master key encryption is valid?
In order to programmatically determine if the current SMK was used to protect the DMK, you should be able to simply attempt an operation that would require the DMK. Such an operation would need to first decrypt the DMK in order to use it. Assuming that you have not opened the DMK explicitly (using the password supplied when creating it), decrypting the DMK will require the SMK. If the current SMK is not the correct SMK, then the DMK won't be automatically decrypted and the operation will fail. So:


If you have a Certificate that is guaranteed to exist in the Database being restored, try using it:

SELECT SIGNBYCERT( CERT_ID( '{certificate_name}' ), 'test' );


That should return a non-NULL VARBINARY value. If the return value is NULL, then the DMK needs to be regenerated (per the instructions below).
If no Certificate is guaranteed to exist in the Database being restored, then try to create one. If the SMK can be used to automatically decrypt the DMK, then the Certificate will be created, else the operation will fail:

BEGIN TRY
  CREATE CERTIFICATE [TestCert] WITH SUBJECT = 'yadda yadda yadda';
  DROP CERTIFICATE [TestCert];
  PRINT 'All good, yo!';
END TRY
BEGIN CATCH
  PRINT ERROR_MESSAGE();
END CATCH;



However, since you just restored a database coming from another instance and did not restore that other instance's SMK into the new instance, it is safe to assume that the answer is: "no, the DMK is not encrypted with this server's SMK."

This is an expected scenario that requires the following steps to remedy:

USE [newly_restored_db];
OPEN MASTER KEY DECRYPTION BY PASSWORD = '{password}'; -- password used to protect DMK
ALTER MASTER KEY REGENERATE WITH ENCRYPTION BY PASSWORD = '{password}';
CLOSE MASTER KEY;


The MSDN page for CREATE MASTER KEY states (emphasis added):


  For SQL Server and Parallel Data Warehouse, the Master Key is typically protected by the Service Master Key and at least one password. In case of the database being physically moved to a different server (log shipping, restoring backup, etc.), the database will contain a copy of the master Key encrypted by the original server Service Master Key (unless this encryption was explicitly removed using ALTER MASTER KEY DDL), and a copy of it encrypted by each password specified during either CREATE MASTER KEY or subsequent ALTER MASTER KEY DDL operations. In order to recover the Master Key, and all the data encrypted using the Master Key as the root in the key hierarchy after the database has been moved, the user will have [to] either use [the] OPEN MASTER KEY statement using one of the password[s] used to protect the Master Key, restore a backup of the Master Key, or restore a backup of the original Service Master Key on the new server.


The MSDN page for OPEN MASTER KEY states:


  When a database is first attached or restored to a new instance of SQL Server, a copy of the database master key (encrypted by the service master key) is not yet stored in the server. You must use the OPEN MASTER KEY statement to decrypt the database master key (DMK). Once the DMK has been decrypted, you have the option of enabling automatic decryption in the future by using the ALTER MASTER KEY REGENERATE statement to provision the server with a copy of the DMK, encrypted with the service master key (SMK). When a database has been upgraded from an earlier version, the DMK should be regenerated to use the newer AES algorithm. For more information about regenerating the DMK, see ALTER MASTER KEY (Transact-SQL). The time required to regenerate the DMK key to upgrade to AES depends upon the number of objects protected by the DMK. Regenerating the DMK key to upgrade to AES is only necessary once, and has no impact on future regenerations as part of a key rotation strategy.


The MSDN page for ALTER MASTER KEY states:


  The REGENERATE option re-creates the database master key and all the keys it protects. The keys are first decrypted with the old master key, and then encrypted with the new master key. This resource-intensive operation should be scheduled during a period of low demand, unless the master key has been compromised.