I raise that concern in part because I once had a client from New York City who referred to the area just east of Ballard as rural. What you're used to affects your perception.
Ballard is a great community. I would advise also looking into Queen Anne or Magnolia. The eastern part of the Lake Washington is beautiful but the yards are smaller. Another great community is Wallingford, a slightly longer distance but the family charm is one many of my clients have enjoyed.
The most important differential in this discussion is the community. Ballard is a younger more "hip" crowd at times and very diverse.
Believing the best for you,
Ballard is the BEST!
For Amazonian's it works great too just because you can take public transit right in to South Lake Union.
Looks like there are Three on the market now at 4 bed 1.5+ bath under 400K.
Ballard is a fine neighborhood, but of course it will depend on your price range and other factors. Are you in Seattle yet? I'd encourage you to select a Realtor early to start sending you listings and sharing with you about the various neighborhoods and your families preferences.
I did a quick search for 4+ bedrooms and 2+ baths and in Ballard only found 2. In Ballard and the surrounding areas found several more starting in the low $400k range. I don't know if this meets your definition of "reasonably priced" but it's a place to start.
Welcome to the NW. Feel free to let me know if I can help.