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. The Airport Van Shuttle is a good way to travel and will drop you off at Faunce Arch, near the corner of Brown Street and Waterman Avenue. Cabs are also available but are more expensive.
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 MBTA or cab.