I know SE16 better than SE1 from a residential point of view, here's the situation.
Find Bermondsey tube station on a map. This is handy since it's on the right line for Canary Wharf, without having to deal with the scrum at London Bridge.
North-West of that point is posh, broadly "Shad Thames", in a fairly good way, but expensive. If you want a classic warehouse conversion and are prepared to up your budget a bit, this is the place to be.
South and South-East of that point is fairly unpleasant up to the shopping centre, but much more affordable. I live here, but then I grew up in Yorkshire so it comes as no shock to me :)
Beyond the shopping centre is like a poor man's Shad Thames, newer buildings that have shot up to replicate warehouse conversions, but aren't. I nearly lived here (and did live in a flat above a shop here for a while). There's nothing really wrong with it but it's rather soul-less at the moment. The cinema, bowling alley, and so on are here.
North-East of Bermondsey Station (the Rotherhithe Peninsula, effectively) is fairly nice, but a bit of a public transport backwater once you get beyond The Mayflower Pub. It doesn't have much 'stuff to do' within it, either (for a young chap - there's a nice park, and a city farm, and, umm). It's more affordable than North-West, but less than South. Unless you want a river view from your window, then it's expensive again. If you want a warehouse with a basement gym etc for substantially less than in Shad Thames, this is the place to go, particularly the East side.
South-West of Bermondsey Tube is slightly split, the North part of it is very much council flats and tower blocks (but relatively good or improving ones), whereas the south half is more little 1980s housing developments where the slightly better off families in the area seem to live, and some flats that trade off making it into the SE1 postcode. It's ok, but unless you're close enough to feel like you're in Borough, probably doesn't meet your needs.
Having lived in SE1 for over 7 years (London Bridge) I would recommend Bermondsey Street north of Long Lane as the best place to live. It's a lot cheaper to go south of long lane and east of Tower Bridge Road, but it's a totally different vibe to the direct area of Bermondsey St. It's not at all tourist-y (unlike South Bank/Clink Street/Borough Market, which used to be great but are now swamped with tourists). Shad Thames and Borough also worth a look though, Shad Thames is more of a "city" vibe with chain restaurants, Borough around the Marshalsea Road area is is great and probably better located than Bermondsey St for access to the city etc...
Most bits of SE1 are safe enough (as safe as anywhere else in London) if you are sensible.
For somewhere "stylish" the Jam Factory might fit the bill, although I'm not sure how cheap that would be. There are various school conversions as well, which tend to be perhaps slightly cheaper than the warehouse style stuff. e.g. http://www.gumtree.com/london/52/25363652.html