T.F. Green Airport (PVD) is located in Warwick, RI, approximately 9.4 miles from Brown, and is served by most national and regional airlines. Cabs are available at the main terminal.
Boston's Logan Airport (BOS) is a good second choice. Boston is 51 miles from Providence, and there are multiple transit options: bus, which leaves directly from Logan, and train, which requires you to get to South Station by public transit or cab.