Does it help? Yes, I think that in some cases (for the right person and the right program), if you do your visit in the right way, it would definitely help. I know of at least one case where an applicant visited after the initial offers were made, but before the official visit weekend (they did not themselves get an offer yet) ended up getting an offer shortly after their visit. And this is for a school that usually only does one round of offers. The person also got to visit the school again during the official open house!
However, I don't know if it's really worth it because the added cost is really high and there's little guarantee that it will make enough of a difference. Like Riley said, I would only do this if I was going to be in town (or really close) for some other reason anyways (visiting a friend, a conference, just passing through, live really close etc.)
Also, I think you have to be the right person and do the visit in the right way for this to even help you. By "right person" I mean that you need to be the type of person that the department just happens to be looking for but for some reason, your application does not reflect you as well. You also have to be already accomplished -- a visit isn't going to magically make your appear more qualified/desirable to the school. A visit can only increase the awareness of excellence that is already there.
By visiting "the right way", I mean that you have to properly organise the visit -- don't just drop in! Contact someone in the admin office of your department and/or your prospective supervisors and let them know, well in advance, that you are planning to be in town for <whatever reason> and hope that you could talk to some profs or students. Usually, the department will set up a schedule for you. When you are visiting, remember your goal is to find out about what this department has to offer you and also to communicate what you have to offer to them. Make it worth their time too, don't just show up to expecting that the token gesture of meeting in person will magically make your application package better.
Not everyone who visits a particular department is going to improve their application package substantially. And not every department you visit will improve your own application package substantially. I think it can help, but only in the right combination of the right person and the right department. Hence, I don't think it's worth it to just simply try to visit as many departments as you can (huge waste of time and money). But if your schedule works out that you will be in or near one of your schools, then go for it!
PS Alternatively, you can try to meet and chat with students and professors from your prospective schools at an upcoming conference! For example, in my field, the January meeting of the American Astronomical Society is an excellent time to talk to people at schools you're interested in -- it's right after you finish your applications and usually right before profs start meeting to discuss the applications! However, I wouldn't do this out-of-pocket unless you live really close! Best case is to present your research and get your supervisor to fund you but another option is to volunteer at meetings, which will sometimes give you reduced or waived registration costs. Also, I would advise you to email people you might want to meet ahead of time, as it might be pretty hard to find specific people in a busy meeting.