Look up national, reputable, well-known "suites hotel" chains as Everly suggested, such as Embassy Suites or others. She's right -- avoid New York Ave., Rhode Island Ave., North Capitol Street; not good areas at night and the so-called family hotels are not in good shape either. I would definitely not stay out in the suburbs such as Chantilly or Reston (both Va.) as others have suggested; you will spend huge chunks of each day slogging back and forth on Metro (our subway) among the commuters, and Metro does not even go to Chantilly or Reston! (You do not, repeat, not want to drive into and out of DC each day; the traffic is second only to LA's as worst in the nation and you will deeply regret not staying in the city. Better to pay more to park and then walk or Metro everywhere than to be cursing the traffic twice daily for, oh, an hour or more each way.) The traffic from the suburbs into the city is infinitely worse now than in 2003 when Malia B. stayed in Chantilly, believe me. We live in those suburbs.
If you just must be outside the city, you want a close-in suburb such as Arlington or Alexandria, both in Va., but again, I'd stay in the city.
Good areas these days include the very hot Penn Quarter area (near the Verizon Center arena and the American Art Museum) and Dupont Circle, though I don't associate Dupont Circle with larger hotels that would have suites and Dupont Circle isn't walking distance from the mall for smaller kids. I would get a very good, current DC map and use it to plot some areas. If you are unsure about safety of an area, you could check Tripadvisor.com reviews for specific hotels, but be aware, those reviews often are only from people wanting to vent about negative experiences and don't always fully reflect an area' s positives.
DC is the best. Go to www.si. edu, the Smithsonian's web site, and search on family events for lots to see and do. The International Spy Museum (not part of the Smithsonian) is great for adults and older kids (maybe 12-14 and up) but will bore younger kids stiff and does cost $$. Do get to Arlington National Cemetery if that interests you, and if you have the time, take a day trip by car (this drive's fine) out to Mount Vernon, the home of George Washington (allot a full day for it or at least a full morning through lunchtime). It's only about 30 minutes by car from downtown DC and an easy drive. Check the Washington Post web site for family shows and activities in the area, and look for children's plays and other theatre. Getting into the White House now is more difficult and I'm not sure how it's done any more but check at their web site. The Capitol has a lovely, huge new visitor's center and worthwhile tours. The Washington Monument, alas, will be closed for the rest of 2012 due to earthquake damage.