The Social Science Research Network has published two new rankings of both schools and individual scholars (this will be released soon) by the total number of downloads their papers have received from the SSRN site. As a measure of scholarly output and quality, this one has many limitations: SSRN, for example, is utilized disproportionately by scholars working in just a few areas (corporate law, law & economics, intellectual property), and it is utilized by some schools (typically those that excel in corporate law and law & economics) more than others. Papers that are "surveys" or "overviews" of an area often tend to have much higher levels of downloads, presumably because they attract both scholars in other fields and students.
The rankings of downloads-per-school are here (you must be an SSRN subscriber to view this); the top 25 in total downloads in the last year (which is how the SSRN data initially appears) are as follows:
1. Harvard University
2. Stanford University
3. University of Chicago
4. Columbia University
5. University of California, Los Angeles
6. University of Texas, Austin
7. George Mason University
8. University of California, Berkeley
9. University of Virginia
10. Yale University
11. George Washington University
12. Georgetown University
13. Vanderbilt University
14. New York University
15. University of San Diego
16. University of Pennsylvania
17. Boston University
18. University of Illinois, Urbana-Champaign
19. University of Michigan, Ann Arbor
20. Fordham University
21. Florida State University
22. University of Bonn
23. Duke University
24. Cornell University
25. Boston College
Note that Larry Solum's move from San Diego to Illinois will surely change the relative rankings of those two schools.
The ranking of downloads-per-scholar will be reposted; it turns out the data on-line earlier today was actually out-of-date. I'll repost on it here when the new data has been processed.
Note that you can reorder the rankings by clicking on the various columns: e.g., you can rank individuals and schools by downloads per paper; by total downloads since the inception of SSRN; and so on. As I understand it, these rankings will also be continuously updated as papers are downloaded.