The landlord usually can't tell tenants who can sleep in which bedroom.

However, I'm wondering if this is a legal bedroom. There are laws and regulations that define what can and can't be used as a bedroom. Those regulations vary from place to place, so you need to research what the laws in your state are.

It is very possible that, in your area, basement bedrooms are not allowed, or that they have to have very specific exits, in case of emergencies. "Access to an egress" is very different from an exit that is in the room. It sounds as if you would have to leave the bedroom and go to a door or window to exit the building. This may or may not be allowable under the building code.

The landlord could easily get into trouble if the law does not allow for tenants sleeping in that room. If it is not a legal bedroom, the landlord has every right to tell you not to sleep in there. She does not have the right to tell you who has to sleep where, though. You could easily turn the living room into a third bedroom, if you wanted to, and use the basement room as a living room.

If the basement bedroom is part of your apartment, could you put most of the non-bedroom things down there, and have the two beds in the same bedroom? It would be awkward, I know, to have to go down to the basement to get clothes and books and things, but it might be a solution to the problem.

One last thought--how on earth did the landlord discover who was sleeping where? Is she entering your apartment without advance notice? Is your roommate talking to her a lot about what goes on in the apartment? I'd want a lot more privacy with my landlord than you seem to have.