What would be the correct form and why? Both could be used, but I think the do version is colloquial. I am interested in the reason though
I hold doors open just to see people do that little run or I hold doors open just to see people doing that little run?
Both are grammatical, but have slightly different meanings.
The one you probably want is do: there the object is the whole clause people do that little run. There's an implied complementiser, such as that or how.
If you use doing, then the object is people [doing that little run], i.e. a noun phrase which contains a participal clause. But the import is that what you are watching is the people, rather than them doing the run. Doing that little run modifies (and probably selects) the "people", but the object is still the people.
