Great answers from all my colleagues!
The main thing that struck me immediately upon reading the question was that In order to encourage employees to actively seek help -- from their managers and from their peers, organizations in general, and the managers specifically, need to create a culture and a climate that not only allows for help-seeking, but encourages it. In too many companies I’ve seen managers SAY they have an “open door policy” (“You know you can always stop by and ask me anything, anytime!”), only to make their people regret it immediately thereafter when the manager makes the person feel stupid, inadequate, or embarrassed for daring to make themselves vulnerable. And once embarrassed in front of their peers, how many employees are going to themselves in that position?
Managers need to make THEMSELVES vulnerable by admitting that THEY don’t have all the answers, and by acknowledging when they make mistakes. Rather than sweeping mistakes under the rug or blaming others, managers who act like leaders find learning opportunities and teachable moments on a daily basis and created a climate of learning and sharing. And they demonstrate collaborate sharing by seeking input and opinions from those around them -- up, down, and across. By modeling the behaviors they desire in others, and practicing what they preach, it will quickly become part of the organizational and departmental culture.
Employees live what they learn. And they learn the most from their direct managers -- as well as from the peers who step up to leadership by facilitating dialogue and debate about how best to solve common business challenges. As a manager, seek to catch people in the act of helping one another, acknowledge and recognize it, and you'll soon find collaboration and cooperation to become a habit practiced by all.