Does AIIMS Delhi sponsor those research trips to Hopkins?

No, the research trips to Hopkins are not sponsored by AIIMS Delhi and the students alone have to bear the expenses of the same.

It’s the students who initiate the process and executive everything without any help from the institute. 

They start by sending their CV to the professor, getting the required medical tests done, booking the tickets, visa and looking for a decent place to stay. On some occasions, AIIMS Delhi even opposes its students going to Hopkins at the expense of missing classes here.

Personally, I feel even if the trip is unsponsored the expenses you bear are worth the result. You are exposed to a foreign work culture and learn so many new things you would never have been able to in India. You also get to explore different places!