question

The problem is, IMHO, not if a full backup was ever taken, but if a full backup EXISTS, and can be found and restored succesfully.

Otherwise your DIFFERENTIAL backup is useless.

But here is how to see the last date a full Database backup was taken:

SELECT 
T1.Name as DatabaseName, 
COALESCE(Convert(varchar(12), MAX(T2.backup_finish_date), 101),'Not Yet Taken') as LastBackUpTaken,
COALESCE(Convert(varchar(12), MAX(T2.user_name), 101),'NA') as UserName
FROM sys.sysdatabases T1 LEFT OUTER JOIN msdb.dbo.backupset T2 
ON T2.database_name = T1.name AND TYPE='D'
GROUP BY T1.Name
ORDER BY T1.Name

Copied from mdsn

I agree with Henrik and Grant. Make sure your full backups are available, not corrupted, and relatively recent (exact definition of relatively recent depends on a number of factors, primarily how fast your data changes) before the thought of differential backups even comes to mind.

With that said, let me point out that some 3rd party tools make this process much simpler than doing it with just SSMS and T-SQL. I personally prefer Red Gate SQL Backup, but several other companies also have good offerings. These tools generally give you an easy to use graph of what databases have been backed up when and how and will write the scripts to automate future backups for you. I cannot speak for the others but Red Gate also has nice features like compression and encryption.