I lived there for a couple of years. Overall, I quite liked it. Yes the town centre is pretty grim / dull and it's not a great place for bars / restaurants but it has lots of green space and housing is reasonably affordable. It's very well served by trains and motorways so it's easy to get far away quickly!
The local riding is actually OK. Heading south there are lots of nice bridleways and back roads for road riding. There's some woodland singletrack (semi-cheeky) if you know where to look. The surrounding area has some really nice country pubs and stuff like that. The bridleways can be a bit grim in winter but I have some great memories of sumemr riding around there.
Oh and it's referred to as either Amizingstoke of Basingrad (reference to the soviet-esque arcitecture!)
Edit: with respect to nice places to live within 10-15 miles. I also lived in Winchester and the surrounding villages and they're very nice (try somewhere like Kingsworthy). There are loads of villages around basingstoke that are nice. Prices are moderate to high I'd say, depending on proximity to transport links. Basingstoke itself is on the cheap side for the South, although there are nice areas.