Footwear will totally depend on conditions.
With no snow on the trail, trail runners would be great, (Note I say trail runners rather than sneakers, since sneakers will be too soft and have less protection against protruding rocks, etc. Trail runners will also be waterproof, but arguably not the most important thing to look at for a summer hike.
In Spring snow, I'd go with a boot to hold your crampons
Fall, with loose, unconsolidated snow, I'd go for more of a boot if there will be drifts, possibly with gaiters.
Winter - full on Mountaineering boot
Sock selection is equally important as footwear...
