I really don't know what they would do with a DNA sample other than study DNA. I don't see how this would affect most people especially when they don't really have to do this when blood tests are so common in medical treatment. Swabs are common to test for a lot of things too, so if you don't want someone to have your DNA, you probably shouldn't get swabbed or have a blood test either.
Stem cells are a different matter because stem cells have the potential to create genetic modifications that the cells from a swab of your saliva don't have. When I was having children, there were a lot of advertisments to collect the cord blood, which would be a way of preserving matching stem cells that could be used in the future in case my children needed some kind of medical remedy this would be useful in producing. The stem cells would already match my babies and they wouldn't have to wait for a matching donor like most people have to do now if they need a new liver or whatever. That rubbed me the wrong way at the time and my children do not have any cord blood stored for later use anywhere. I don't know how they will feel about this one day, but that is what I felt was the best decision at the time.
I haven't heard much about this since and I don't even know how present this subject is if you are having a baby now. My baby is almost a teenager. Maybe it didn't get as much interest as people thought it would, and it is not much of a thing anymore. Either way, collecting stem cells seems much more concerning. The only thing they could do with a DNA sample is study how a person is developed, but they would need some other mechanism to influence this process. DNA is like closed source programming. You can't just hack into it and make changes any easier than you could hack into the source code for Java and make changes.