I definitely wouldn't vote for a system that makes people submit their DNA to the government.
The government can't be trusted to keep my national insurance number to itself, so I most certainly wouldn't entrust it with the details of my DNA. Can you imagine how much a DNA profile of everyone in the UK would be worth to, say, a health insurance company.
I don't trust the government's security measures enough to give them my DNA.
I'd like it if my Son had his DNA on record, it would be another way for me to look after him. If he went missing, the police would be able to tell if he had been somewhere. I'd like to have a chip in his mobile phone (if he had one) so I could tell if he was going to places that I'd rather he didn't. Further to that, I'd put a GPS chip in his neck so I could track him with my satnav. If I could. When he's old enough, I'll log all of his msn chats and check his internets history. I want to look after him and keep him safe, make sure that the decisions he's making are the right ones according to my values.
I don't want the government doing that to me. DNA records being kept for people without criminal records is a step in that direction.