I work in Southbourne, next to Boscombe, and travel through the latter every day. Although it's true that it is being tidied up a bit, I really don't think I would choose to live there. There are areas which have a reputation for being 'druggy-ville' and some places you really wouldn't want to walk around on your own after the shops have shut!
Springbourne, which is close by (but further from the sea) is a reasonably-priced area with a better reputation; Southbourne is pleasant, although there have been a few racist attacks on foreign students recently; and, as Cetti says, Boscombe East is much nicer - which makes it more expensive.
Good luck with your house-hunting.