Something really interesting just happened in the registered demographics of the United States. In the past decade, we saw a shift in the trend of people moving away from major urban centers (those metropolitan areas with an average population of 1 million or more). Since the 1950's the process of urbanization has had a strong wing in suburbanization, the creation of population centers that are not rural but are also fairly distributed compared the the inner city. As far as where people are moving in 21st century America, the major urban centers are getting an influx bigger than the surrounding suburbs. There are a lot of reasons for this, economic issues being chief among them. It's an entire lifestyle change and it's forcing people to address concerns society as a whole has been avoiding for decades. One such concern is the reality of raising children in a major urban area when several generations have promoted a preference for a suburban flight when kids come into the picture. In that sense, Seattle is an unusual place. Our "suburban" neighborhoods are distributed throughout the city, sometimes just a couple blocks from significant urban buildup. This has made the nature of raising a family in Seattle break the old city vs. suburb mold.
Because Seattle has had a history of mayors and legislators who have been especially careful about real estate zoning, much of the greater Seattle area presents an uncommon mix of business, industrial and residential space representative of a wide income spectrum. While it's kind of a joke among locals, there's a lot of politics behind why the modest housing in central Capitol Hill bumps up against the large, lavish homes of the neighborhood people have come to call "Millionaire Row". This sort of arrangement is fairly common around Seattle, the generally lower income housing of younger residents and lower middle class workers sharing cross streets with the residences of the firmly established family, as well as local businesses along main thoroughfares. Suburbs, in the traditional sense, exist in the downtown core itself.
Even outside of downtown there's remarkably little division between the high and low end of the socioeconomic scale. Take the area around Green Lake, for instance. Much of the Green Lake neighborhood is composed of upper middle class family homes, maintained parks, schools and libraries stretching all the way to the University District. But along the west edge of Green Lake, Aurora Avenue North stretches in a strip of greasy spoon restaurants, cheap motels and at least one adult video store, reaching past the gentrified center of Northgate Mall to a patchwork of wooded suburbia, aging shopping centers, low stakes casinos and the Edmonds highway exchange. But then, just east of Aurora is the decidedly family-friendly Greenwood/Phinney neighborhood, quiet suburban homes lining every street up both sides of the ridge.
In order to find true suburb in the Seattle area, neighborhoods that don't bump up against the hallmarks of the urban center, one has to stretch beyond the Seattle zip code. By name and in spirit these areas aren't really Seattle anymore. If what you want is Seattle proper, there are plenty of options for the family lifestyle, it'll just be an experience akin to raising kids in some hybrid of Brooklyn, New York and Beaver Creek, Oregon. This can be a great opportunity, provided you don't mind swallowing the somewhat steep real estate prices here. Kids can get the best of both worlds, the vibrant city of culture and the quiet home in a safe neighborhood. Truth be told, it'll be harder to control the flow of information, so it won't be entirely up to you when your child learns about some of the more complicated topics (can you say "drag show at Julia's on Broadway"?), but this is the age of the Internet, so it's best to be realistic about how easily we can censor our kids' experiences anyway.
Lastly, there's the matter of schools. Raise your kids in Seattle and you'll have a lot to choose from, at least in the private sector. Seattle public schools are decent but don't really get a lot of local support. A pretty significant portion of the student population in Seattle is educated at smaller niche schools, from the standard religious private school (pick a major world religion and we've got a private school for it), to tech schools and alternative programs of every stripe. Expect to do some shopping around and try to be smart by getting your name on at least a few lists well in advance of enrollment. This model is pretty standard for most major urban areas these days. It may not be the iconic image of all the neighborhood kids piling into a yellow bus to go to the same school, but if you've chosen to live in Seattle that kind of conformity is likely far from your preferences anyway.