Yes, they do and those checks are carried out strictly. Indian nationals are required to have a photo ID as proof and foreign nationals need their passport and visa as required by the nationals of the country they belong to.Also, do they have border checks to check if someone is Indian or Foreign?
This is very interesting and I should read up on visa requirements. I would love to go to Nepal while I’m in India. But it is requires specials visas and problems getting back into India, I might change my mind.Indian citizens do not need a visa to enter Nepal and there is also a direct bus that goes to Kathmandu from New Delhi that might be of interest to you.
You can go through this post about Nepal visa for foreign nationals.