I don't think the Anne Frank House is appropriate for young children, and not just because of the obvious hurdle of explaining war, Nazis and concentration camps to them. The museum can feel a bit cramped, it's often crowded and it evokes a somber mood -- it's not a place where it's appropriate for a young child to be darting around. Pre-school-aged children would likely get bored and antsy, as the areas are small and relatively empty, except for photographs (some graphic), preserved objects and written descriptions to help visitors envision the original conditions. I have seen children as young as 7 years old there; they seemed to be understanding the museum with explanations from accompanying parents. But to be perfectly honest, I would say 10 years old is the appropriate minimum age.
On a lighter note, if two parents/guardians are with one too-young child and another they feel should visit the Anne Frank House, there's a fun place to get traditional Dutch pancakes just down the Prinsengracht canal from the museum (Pancake Bakery) -- the group could easily split up and have different experiences, as a visit to the museum usually takes an hour or less.