In reply to AnnaSpanna:
I used to live in a remote rural village about half an hour's walk from the railway halt. My 17 year old step daughter used to catch the train to college and walk to and from the halt across fields. She had been brought up in the country and felt comfortable and at home in it.
When the winter evenings started to draw in, I suggested that I meet her at the halt to walk back with her, but she insisted that she would be fine on her own saying that the chances of there being anyone stalking the country fields at night intent on finding a lone female to molest were negigible. On the other hand she felt a much greater risk walking through the town from college to the station in the evening, and on a couple of occasions had uncomfortable encounters with wierdos trying to chat her up!
I think you will be fine on the South Downs Way, the chances are there will be quite a lot of dog walkers, cyclists and other hikers up there, particularly at a weekend. If you feel uncomfortable aim to be down well before dusk (although for the reason stated above night is probably the safest so long as you avoid the few downland car parks which attract courting couples and voyeurs!!).
BTW I think you will find it hard to do the whole walk over just a weekend. I did it quite a few years ago with two of my kids and we took 4 days averaging about 20 miles a day. We actually wild camped pitching the tent after dark and having it down at dawn. We never saw a soul up there overnight. Although you could see the lights of the coastal towns below it felt really wild and remote - like being up in the mountains!
By far the biggest problem in walking the South Downs Way is finding water. You need to take plenty and fill up at every opportunity.