If you are interested in architecture or you are looking for something to do in downtown San Jose, then a visit to Cathedral Basilica of St. Joseph is the thing to do. The cathedral is listed as a California Historical Landmark and as well as the National Register of Historic Places.
I arrived on a Thursday afternoon and found very few visitors, which made it easy to capture photos of the architecture with out people.
The first church on this location was called San Jose de Guadalupe built in 1803. Earthquakes and fires took their toll of the original and several successive buildings. The existing structure was consecrated in 1877. In 1985 following a major renovation it was elevated to the status of cathedral and made a minor basilica by Pope John Paul II in 1997. The church is open to the public with no fee, although you may want to pay attention to the events happening at the church since it is an operating church with worship services and other events.