I'm not sure about that area specifically (I live in Elephant&Castle) but the area as a whole is an absolute joy to live in. You'll be close enough to Central to be able to enjoy with ease, but far enough away to avoid it if you want to. Always events going on in the immediate Locale to visit, a good night life, good pubs&restaurants and all within walking distance and, above all, a (generally) fabulous mix of residents.
As with every area in London, you need to keep your wits about you and be sensible when alone, but I wouldn't live anywhere else. I feel safe here, which is the most important thing for me (as a 30yr old female who more often than not walks home alone at all hours.)
The area is great - 5 mins walk north to the River at Shad Thames, 1 min walk to Bermondsey Spa Park and less than 10 minutes walk to Bermondsey tube. Plenty of buses on Jamaica Road so transport links are not a problem.
I have lived near the Neckinger estate for more than 6 years and love the area. We are not short of dining options within a short walk. It is less than 10 minutes walk to Bermondsey Square and Bermondsey Street, 2 minutes walk to Cafe Amisha (great pizzeria restaurant on Grange Road), less than 10 mins walk to the restaurants in Shad Thames (Tentazioni, Pont de la Tour, Chop House, etc).
I am fortunate to have many friends in SE1 too - this community website is a great place to meet friendly people.
I live 5 minutes from the Neckinger Estate, regularly walk through it on my way home in the evenings, and store my bike in one of the lockers there.
Never seen any problems or felt unsafe walking through late at night. Plenty of kids around playing in the playground and generally running around, but nothing that has ever worried me. It seems fairly quiet.
As has been said, great location for nearby Bermondsey Street, you also have the market on Dockley Road on Saturdays, a short walk to the Jubilee line to get you to the west end in 10 minutes/Canary Wharf in 5, and lots of friendly local pubs. The blue is also decent for any basic household shopping you need to do, and you've got the big Tesco and an Argos around 15 minutes walk away on Old Kent Road.
Neckinger is a very safe and family friendly estate. My kid's childminder lived there and it's very well kept and often has nice friendly kids playing outside. It's also right next to the wonderful market stalls for Maltby Street market on a Saturday.