Regardless of what you read online, Westcott--which is the hippest neighborhood in Syracuse--isn't actually that hip and cool. It's a rundown strip of like two blocks with an overpriced Mexican food restaurant, a Papa Johns, and a Subway. Everyone raves about it (likely because there's nothing to do in Syracuse except Destiny USA, the mall...but even that), but it's nothing special. As for undergrad issues, you'd be fine in Westcott. It's a mixture of undergrads, grad students, and families. Parties, at least what I can see because I don't party, seem to be concentrated at the greek houses.
If you have a car or plan on riding the bus, try Clarendon Heights. Renovated apts...and unless you have rude neighbors, it'll be quiet. Out of the way, but still in the mix of things!