RMAN list archivelogs that are needed for to recover specified backup I've taken backup of datafiles in 31.12.2013. It completed datafile backups successfully, but archivelog backups failed. 

Now, I want to determine what archivelogs should I backup to be sure that my database will be recoverable after restoring 31.12.2013 backup of datafiles.
Depending on your requirements and what other backups you have done since then, you could use (from http://docs.oracle.com/cd/B28359_01/backup.111/b28270/rcmreprt.htm#BRADV89594):

REPORT NEED BACKUP RECOVERY WINDOW OF n DAYS
Displays objects requiring backup to satisfy a recovery window-based retention policy.

REPORT NEED BACKUP REDUNDANCY n
Displays objects requiring backup to satisfy a redundancy-based retention policy.

REPORT NEED BACKUP DAYS n
Displays files that require more than n days' worth of archived redo log files for recovery.

REPORT NEED BACKUP INCREMENTAL n
Displays files that require application of more than n incremental backups for recovery.


These will tell you which files you need to backup to provide 'n' days worth of restore capability or 'n' redundancy.