I recently applied for a job and I filled out everything online. There is one question that says, "Can we contact your current employer?" I marked, "No," because I did not want them to call my supervisor because I don't want him to know I am applying for jobs. Does doing this hurt a candidate's chances of getting the job? (I could explain the problem in the interview.)
From my experience it is normal to ask that your current employer not be contacted about your job search. Every job I have applied for a job, I have responded with a no to this question. You will not look like you have something to hide by saying no - you will look like someone who doesn't want to put your current employment in jeopardy.
The majority of companies should understand that candidates don’t want their current employer tipped off to the fact they are searching for a job. That being said I'm sure there are a few cases out there were saying no has gone against someone but I've yet to hear of it based solely on this reason alone.
Does doing this hurt a candidate's chances of getting the job?
It depends, many people don't want their current employer aware of the fact that they are applying for jobs. So I think that is understood. You can perhaps alleviate any possible misunderstandings of your choice to deny access to your current employer by allowing them to contact previous employers. Then it should be ok.
However, if your current job is the primary job in the field that you are applying for, then it will be difficult for Human Resources to verify your employment history. If that's the case, then you may need some outside verification of your skills: Stack Overflow, LinkedIn Recommendations, Personal Blog, etc.