Yes, candidates should be responsible for the actions of their staffers. While staffers have their own free will to do what they want, candidates should intensely screen their staffers and encourage good behavior from them. If the staffers do something out of line, the candidate should accept responsibility and make restitution.
Candidates should be responsible for the actions of their staffers because they play a key role in their recruitment and any wrong doing should be shouldered by them. They should ensure the recruited staffers are people with high integrity who poses unquestionable character evidenced by their untainted record in their personal and public lives.
Political candidates should take responsibility for many things during their campaigns. However, no candidate can be held accountable for the actions of her staffers. Yes, the candidate should take action when a staffer does something wrong; maybe even firing that staffer. Candidates can not control what their staffers do: no human can completely control the actions of another human. Therefore, it is wrong to blame a candidate for the mistakes of her staffers.
No, candidates should not be responsible for the actions of their staffers if they had no reason to know that their staffers who commit these actions. If the candidate did not know the staffer personally, the person who hired the staffer is responsible for that person being on the payroll.