travelsafar - before moving to a rural farm in the SW I lived in SE London, 18 mins away from Victoria on an overground train line in a very desirable area. Theft of property (catalytic converters on cars) was routine but attacks on the person not. I travelled with a walking stick regularly on the bus to the tube station in Brixton, people always offered a seat, and I used the tube without problem even in rush hour. But and this is a big but, I was familiar with my routes and I knew the tube stations that had lifts etc. I was never afraid, even on the last tube back. Nobody ever took advantage of me.
Now, with a worsening arthritic knee and having been away from the big city a couple of years, when I do make a visit, I book special assistance (free) on the train and I get help getting on, and when I get off there's a buggy waiting for me with a very professional but also friendly staff member and that takes me directly to the black cab rank because my 'disability' means I don't feel I want to charge around tube stations. Yes, I do feel vulnerable but I want to emphasise that I only feel so because I can't trust my knee not to give way and not because I think London is a hotbed of violent crime. I am definitely with those who believe there's a hype about the dangers of London. More Londoners will help you if you need it than you might imagine. And by the way, I'm actually from the North West where I was taught that Southerners wouldn't give you the time of day - not true is my experience. Also, London is totally multi-cultural and I often think fear stems from ignorance. Many times I've been given a seat or asked if I've needed help walking up a short flight of stairs in a tube by a person who doesn't look or sound like me.
Don't be afraid but include taxis in your budget. Good luck.