Prof. Ruth Simmons, President, Brown University, is quite candid in this Mint interview (with Aparna Kalra) about why "major universities will not [open campuses in India] to a very significant degree":
... [For] the most part, if you are in another country, they (the students) don’t want a second-class programme. If you come, they want a first-class programme, which means they want the same faculty that you have back home, they want the same course material, they want everything to be the same.
By and large, it is very difficult for universities to replicate what they have in their own countries. If you ask your faculty to travel back and forth, that’s pretty imperfect, often faculty don’t want to do that. You end up with faculty who are not vetted in the same way that your campus faculty is vetted.
What, then, makes sense for a foreign university with an interest in having a presence in India?
The answer is: Partnership and collaboration. Here's Prof. Simmons, again:
It is very important when you come into another country to demonstrate respect for the educational system in that country, to demonstrate equality of standards. I think for the most part, people will still want to have collaborations. You know, collaborations are wonderful because that sense of equality is very strong in collaborations.
Your students come to us, our students come to you. Your faculty are engaged in research on this project and so are ours, and that equality is very apparent. It is much harder to do that trying to set up an entire programme unless it grows out of collaboration.
Kalra, the interviewer, mentions Brown's partnership program with St. Stephen's since 1991. Prof. Simmons says she's keen on expanding this program, as well as on collaboration with other institutions.