While giving my answer to another question I realized that Tank and Dozer are on watch reading the Matrix from the Nebuchadnezzar or someone else in the crew is always watching. Other than the Agents though what are they watching out for? Especially when plugged into the Matrix.
There are a number of other things that operators would be watching for:
Programs: sentient machines that appear as human in the matrix and are responsible for performing various maintenance roles. Agents are merely a subset of these programs. Other examples are Rama Kandra (a power plant systems manager for recycling operations) and his wife Kamala (an interactive software programmer). Operators would want to look out for these programs because they could still pose a direct or indirect threat to redpills if they discover who they are or what they're doing.
Exiles: programs that are obsolete, have no purpose, or no longer function properly, but who refuse to return to the Source. These rogue programs are no longer working with the machines, but can be a danger in and of themselves and must be watched out for. Many appear as human, such as the Merovingian, his wife Persephone, the Trainman, the Keymaker, and Sati. Others have the appearance or characteristics of paranormal creatures, such as the Twins (ghosts), Seraph (a wingless angel), and two of the Merovingian's thugs Cain (a vampire) and Abel (a werewolf).
Déjà vus (or generally any glitches): A déjà vu is an indicator that the code of the Matrix has been altered. In the scene in the Lafayette Hotel where Neo sees a black cat walk by twice, the Matrix is changed to seal off the hotel's exits. It appears that this change is simultaneously discovered both by Mouse (in the Matrix) and Tank (outside the Matrix), suggesting that glitches like a déjà vu can still be noticed by operators.