First, you ought to be aware that the Smithsonian isn't A museum -- it's a large number of museums, each of which is huge. Here's a link to their main website:
http://www.si.edu/ I think it listed fourteen museums.
You might want to start with the Smithsonian Castle, which was the first museum. The story of how the Smithsonian was started is interesting.
Most of the other museums are clustered in the same area, but some others are quite far away -- the National Zoo, for example, is part of the Smithsonian. The Smithsonian includes a botanical garden, a sculpture garden, and more! Similarly, the National Air & Space Museum has two buildings -- one there in the center of DC, the other at Dulles Airport!
The Smithsonian owns so much "stuff" that it can only display 2% of it at any one time.
Not every DC Museum is part of the Smithsonian. The Holocaust Museum is not part of the Smitsonian, nor are the Spy Museum and the News Museum (those last two charge a fee for entrance).
Soooo . . . I'd suggest that you decide which museums you want to visit, then find a hotel. Someone else's suggestion about finding one near a Metro station is a good idea. The Metro does stop right there in the middle of the complex.
There's a Holiday Inn very near Air & Space.
I've stayed at the Pheonix Park Hotel, an historic hotel walking distance to the Smithsonian.
I've also stayed at the Residence Inn on Vermont Avenue, which was near MacPherson Metro Station.