Specific US embassy may or may not work during the shutdown. Each individual US embassy maintains a buffer to keep their services moving even in the federal govt. shutdown. If the specific US embassy like the Delhi US consulate, does not have sufficient funds to keep them going, they will inform this on their website specifically. If open, the US embassy also takes interviews as scheduled in foreign locations like India, China, and consular processing happens for H1B and other visas. The government shutdown, normally, is resolved within days, and hence, no operations are affected. In the last government shutdown, all US embassies were open. You should check with the specific US embassy that you plan to visit and see if they are closed or open during the government shutdown.