The things is that Basingstoke as a town is pretty soulless, I'd hate having that as my home town (sorry to all Basingstokites!).
Winchester has been mentioned above. It is posh and twee and pretty, and not at all ethically/culturally diverse. It's solidly middle class, which as its pros and cons. Also very expensive. Some nice villages surrounding it though.
Eastleigh is its working class neighbouring town, which has good schools too (Thronedn, Toynbee) and is affordable by comparison (just not posh and twee
) and the snobby Winchester people call it Beastleigh. It is not actually that bad!
The Comprehensive system seems to work in Hampshire and is seen by many as a good thing.
Perrins in Arlesford and Thornden are also very good, I have heard. There must be more good schools, but those are the ones I know.
My DS is at a Winchester Comp, and the great thing is that sets are flexible, and he has moved up from middle set to higher set through working hard (and from lower to middle for English, he's dyslexic). If you have a grammar system, you cannot "work your way up" like this. Similarly, the lower sets have smaller classes and extra staff. Think it is one of the better systems in the UK, IMO. Most comps here get around 80% 5A*-C incl English and Maths and most go on to do A-levels at Peter Symmonds or Barton Peveril (both excellent).
Winchester and surrounding area has lots of good primaries, and when we moved we just contacted one directly and were given a place before we had even moved (DS was in yr 1 already so it was up to the school).