If you are in India on a research visa, but are not sponsored by USIEF (Fulbright), you are going to need a notarized letter from the U. S. Embassy in order to access the National Archives of India in Delhi.* Contrary to the many rumors circulating on the The Internets, the U.S. Embassy does provide this letter for a $30 fee as part of the notarial services they provide to U.S. citizens.
To receive this letter, you should take the appropriate ID (passport, visa, letters from home university, etc.) and money to the American Citizens Services Unit at the U.S. Embassy in New Delhi. The Embassy is open for routine consular services from 9:00 a.m. to 1:00 p.m., Monday through Friday. The Embassy closes on a surprising number of holidays, most of which you've probably never celebrated in your life, so be sure and check the list of holiday closings before you head over there. You don't need an appointment, as it's first come, first serve.
To find the American Citizen Services Unit, go to Gate 6 of the Embassy (generally known as the Visa Gate) located on Nyaya Marg in the Chanakyapuri neighborhood. Nyaya Marg runs parallel to Shanti Path, to the west. If you can find Panchsheel Marg, that runs perpendicular to both, crossing both Shanti Path and Panchsheel Marg to the north of the Embassy. Another relatively close landmark is Nehru Park--it runs parallel to Shanti Path along Niti Marg, to the east of Shanti Path. I think the closest Embassy to the U.S. Embassy is Finland's, so if you see that, keep going, you're almost there.
*If you on a Fulbright-Hays DDRA, USIEF will arrange for this letter, free of charge. It takes about a week, so if you're in a hurry, you could always follow the above advice and get the letter yourself (for a fee).