babylon

Cornell. Went there undergrad and my advisor was a China scholar; I can attest to their strength in that area. Ithaca is a great place to live (depends on the person, but I loved it). PM me if you have any questions.

I would agree with @VMcJ - Cornell is a very reputable program, if you don't mind living in Ithaca. There's also no guarantee what will happen after you get your MA, since the variables all change for each cycle (who applies, how competitive the round is, whether funding gets cut for programs, etc.). I actually know someone who turned down a PhD to do a Master's at his ideal institution, hoping to get into the PhD program afterwards, and was unfortunately not accepted to the PhD at his Master's institution.
The only thing that would convince me to recommend the MA offer is 1) if it is fully funded (cost of living in NYC is extremely high), 2) if you think you can gain some substantial training in it and some career clarity, and 3) if you would be happy with an MA if you were not accepted into your preferred PhD program in the following cycle. 

Cornell ftw

Fully funded PhD offer at an R1 vs. a non-funded MA, isn't it obvious?

If you (1) feel that you can effectively address the weaknesses of your application until next cycle (- and doing well in methods/research design classes at Columbia will certainly add to the quality of your application), (2) have a strong preference for working at a top 15-20 university after finishing your PhD, and (3) can afford the MA at Columbia, I could see why you are considering to turn down Cornell in order to get into a better program in the future .
There is an undeniable difference in placements between, for instance, Harvard/Stanford/Princeton and universities in the top 8 to top 12 range. However, only you have the information to calculate whether the expression (expected benefit of getting into a better program*probability of getting into a better program + advantages of the MA at Columbia (network, knowledge, prestige,...)) outweighs the costs of reapplying (MA tuition/costs of living, probably entering the job market one year later, risk of not getting into a program at all,...).
I am in a similar position: I have a fully funded offer from a top 10 university, but I am not entirely happy about this cycle and I am currently trying to get more information on what my chances would be if I were to reapply next year (with a LoR from a methods Prof., more field experience, better GRE)