The award is the oldest from the Harvard theatrical group.
The Baltimore theatre recently postponed a planned production.
Signature Theatre in Virginia will present the award in April.
Brewer is currently a producer at the theatre.
The San Francisco and Oakland locations will both close March 18.
The fellowship comes with an unrestricted $50,000 cash award, accompanied by a year of financial planning and additional artist services.
After 25 years in Minneapolis at a theatre he co-founded, Rothstein will succeed longtime artistic director Michael Donald Edwards at Asolo.
The theatre’s new building has been recognized by the American Institute of Architects and USITT.
Smith takes up the newly created position after working at the Manhattan School of Music and New York City Opera.
Awardees include Felicia Curry, Kate Eastwood Norris, Lawrence Redmond, and Helen Huang.