Arrive early. Standard protocol advises an arrival of at least an hour to an hour and a half for travel with identification. Many airlines allow check-in three hours before departure. When arriving without identification, check-in at this time, or at the latest, 2-1/2 hours, would be prudent.
Go to the appropriate airline counter. Do not attempt to check-in at a kiosk, secure advance online check-in or otherwise prepare for your flight without speaking with a representative.
Explain that you have lost your ID or had it stolen. Some airlines make a notation on the itineraries of individuals traveling without proper ID or at the very least, on the boarding pass.
Go to the security checkpoint. When prompted for your identification, explain what has happened. Do not simply say you don't have it or refuse to show it. You must state that your identification has been lost or stolen. The difference between refusal to show an ID to the Transportation Security Administration and having lost an ID is the difference between boarding and not boarding. Now is not the time for privacy.
Submit to secondary search. Once at the security checkpoint you will be asked to step aside after indicating that you have no additional ID. Either near the checkpoint or in a private room, a pat down, wand search and search through your check-in luggage will take place. Once cleared, you will be allowed to board your flight.