I'm in downtown Seattle as I type!
My office is in downtown directly across from Pike Market. I agree with PPs who recommend avoiding Pioneer Square with kiddos, at least for your hotel. I also like Silver Cloud right by the Stadiums. I've never stayed there, but I checked it out for the possibility of hosting a work function and it's beautiful inside. Also, random fact: Jimmy's on 1st, the sports bar in the lobby, has one of the best burgers in Seattle. The location is great, too.
If you opt to be a little further north, I'd recommend the Homewood as well. It's a bit of a distance from most the "touristy" sites, but super convenient. Unfortunately, there aren't a ton of affordable convenient hotels here. Most the centrally located spots are either local boutique style hotels or high end national chains (W, Four Seasons, Fairmont, etc).
While you're here there will be a really amazing exhibit at the Pacific Science Center/Experience Music Project. It's a Star Wars exhibit! If you or your family are into SW at all, I'd add that to your "must do" list. It's not very expensive and is located literally at the base of the Space Needle.
The only other consideration is this: if you're coming in late July, you'll be here during Seafair, which is the largest multi-week event in Seattle each year. The Blue Angels come, there's a marathon, hydro boat races, parade, etc. If you're coming during Seafair, your hotels will be mega expensive and hard to come by. If that is the case, I recommend staying in Southcenter (near the Airport) and using light rail to get in and out of downtown. That will save you insane amounts of money on lodging. There's a Homewood there, as well as a Fairfield, Embassy Suites, Hilton and a few others as well. Also a newly renovated monster mall with tons of restaurants.
Sorry for the long post.

Happy hunting, and let me know if you have any specific questions.