My information comes from the Washington Post and personal observation. The barricade fences are already staged in preparation for the shutdown. However, as djp98374 points out, there are private museums in Washington. The Kennedy Center will remain open as it is a revenue producing entity and would cost more to close than to keep it open. Most of the other privately funded memorials/parks to which djp98374 are not in Washington, but in other parts of the country.
The key thing is not to assume that your planned activities will go on as expected. Make the telephone call first to find out if the museum, monument, park, or event which you planned to attend is open/taking place as scheduled. Washington can still be a good place to visit, even with a fed shutdown, but if you have your heart set on going to Air and Space, better to find out earlier than later.