Amtrak is great! Takes you from NY Penn Station to Union Station in DC in about 3-4 hours for $49/each way if you book in advance.
Or, you could take the bus from NY-DC. Washington Deluxe, Bolt Bus, and a few others. If I'm late booking Amtrak deals, I'll book the Washington Deluxe bus. Very safe, pretty comfortable, free wi-fi, restroom on board for only $40 roundtrip, some other DC-NY buses are $35 roundtrip.
Hotels in DC are mostly centrally located and safe. Anything downtown or with a NW (Northwest DC) address is very likely safe. It depends on your budget and if you prefer boutiques, large hotels, or budget. Check travelocity, expedia, orbitz and hotels.com to compare the best deals. Lots of families like the Grand Hyatt. Great location, right above a metro station (and Macy's). Plenty of food and shopping in the neighborhood.
As far as things to do in DC, there are plenty. Smithsonian museums are free and very nice. Kids usually love the American History Museum (free) and the National Mall. Just be warned that there's no food near the national mall, eat lunch just before going or pack light snacks and water.
Kids also love the International Spy Museum (about $15, I believe) and the Newseum.
let me know if you have any ?