Apologies, are not a sign of weakness. In fact it takes courage and confidence to acknowledge you did something to warrant an apology. Humans make mistakes. Admission of error(s) acknowledges that you are taking responsibility for your actions. This demonstrates to others(employees, children etc.) good role modeling. It is how we handle our mistakes that define us. Individuals who do not admit their miscues, could run into problems in life areas, such as work, social networks, and family.
However, someone who apologizes frequently, could be viewed as being insecure. Frequent apologies not only are annoying, but they lack sincerity after awhile. In situations that individuals are apologizing for situations that are clearly not their fault or do not warrant an apology, are unsure about themselves and/or the situation they are in.
Also, Individuals who keep apologizing for making the same mistake(s) over and over lack credibility or confidence they are in control of the situation. The people around them see them as not wanting or able to change. This could be construed by someone as weakness or inability to master a situation. There should be a healthy balance between taking ownership of mistakes and apologizing frequently or unnecessarily.