Life in the United States: Procedures
If your passport was stolen, the first thing you should do is to file a police report. If you lost it, then you don't need to tell the police. In either case you should call the Stamford office to speak to the Compliance Officer. Then you should contact the consulate (or embassy in DC) of your home country and get the process started to replace your passport.
If you lose your passport, you have also lost your visa. To replace your visa, you must return to your home country once you have received a new passport and go back to the U.S. embassy and apply for another visa. You must go in person. There is no guarantee that you will receive another visa, so this is a big risk. With no visa, you cannot travel outside the U.S., but you can stay in the U.S. to finish the program.
It is best to use a different form of identification and leave your passport safely at home. Make a copy of your passport and visa and keep it in a safe place.