Yes, New Alresford is a safe area, and comfortably so. The recorded crime rate runs at around 31 offences per thousand people a year. That sits roughly 52 percent below the average for the wider Winchester district. Across a full year the town logged something in the region of 260 recorded crimes in total. For a place of a few thousand people, that is about as quiet as English market towns get.
Both of those are published figures rather than something to reverse-engineer at the kitchen table, and they point the same way. A rate of about 31 per thousand against a district average closer to 65 is not a rounding difference, it is roughly half. What that means in daily life is a town where the kind of crime that shapes where people feel comfortable walking is uncommon rather than a running concern. The honest way to put the town's relationship with danger is that it peaked a long time ago and had nothing to do with people. New Alresford made its name as a hazardous place to live back in the seventeenth century, and the hazard was fire, with a run of blazes that razed large parts of it in 1689, again in 1710, and once more in 1736. The nearest it came to a battle was in 1644, when the fighting on Cheriton Down reached the outskirts. Three and a half centuries on, that old history is still more eventful than anything in the current crime record, which tells you most of what you need to know.
On the numbers, this is one of the safest places you could reasonably pick anywhere in the region. Buy in New Alresford and personal safety is not the variable you will spend any time worrying about.