This question, initially raised by Cain after he had killed his brother Abel, represents one of the historic issues of all time. What is my responsibility toward my fellowmen? Interestingly or perhaps frustratingly, God does not directly answer Cain. So, at first glance, we appear unable to know God's response. What do we assume was God's answer? Yes. No. Maybe. As the murderer of his brother, we do know that Cain readily assumed the role of judge, jury, and executioner of his brother. In that role was Cain his brother's keeper?
What does the Bible say, in general, about whether we are our brother's keeper? Like just about any subject you want to choose, a bible verse here and there can be found to support the answer I want to believe. Paul says in II Thessalonians 3:10 that those who do not work should not eat, apparently tasking each person with the responsibility for their own welfare. Perhaps a different lesson would come from the mouth of Jesus when he taught that divine blessing accrued to those who were themselves a blessing to others, especially in giving and sharing from their abundance.
We can certainly make a case that by promoting independence in others and not unduly supporting their every need, we actually help them emotionally. I think the essential element which determines whether such withholding of support is valid is the matter of motivation. Stressing personal responsibility and perseverance, can stimulate emotional growth and a better self image, but there are very powerful and subtle factors which greatly influence each individuals ability to function independently. Additionally, a part of life's lessons is the realization that we are not truly independent of each other and of the need for mutual support in achieving success. If we are moved by a genuine concern for a person's ultimate good and for that reason force them to accept some personal responsibility, that can be ultimately benevolent. If, however, we are motivate merely by selfishness in forcing others to fend for themselves, then that same action does not seem noble. The intent of the heart forms a very fine line indeed.
Perhaps the answer to the question is yes and no. We do well to take charge of our lives whenever we can, directing our own destiny, hopefully in the proper direction, given our more intimate knowledge of our dreams, hopes, and talents. At the same time, we must recognize that independence to the extreme is a denial of the essence of being human, the need for relationship, community, and a measure of self denial in the interest of others.