Take the BART to the Downtown Berkeley station, then walk across the street to the area immediately in front of the Bank of America, where you can catch the UCB "P", "C" or "R" bus to the Hearst Mining Circle on campus, and there transfer to the"H" (Hill) bus to MSRI. For schedules please visit: http://pt.berkeley.edu/around/transit/routes. Bus travel time from Berkeley BART to MSRI is typically 30-45 minutes, and can be more during the middle of the day. The shuttle schedules change from time to time, and the bus does not run on weekends and UC Berkeley holidays. For current information, visit UCB Parking and Transportation.
On Saturday, Sunday and holidays you can use the AC transit service, bus #65 between Berkeley BART and Lawrence Hall of Science via Euclid Ave. and Grizzly Peak Blvd. There is an hourly service Saturday and Sunday, and a half hour service weekdays (Eastbound from Bart to MSRI, and Westbound from MSRI to Bart). You can find a link to the schedule here: http://www.actransit.org/maps/schedule_results.php?version_id=3&quick_line=65#schedule
Visit http://www.actransit.org for AC transit maps and trip planner.