There is no free parking in downtown San Jose, but there are many parking lots in downtown San Jose, and most of them cost less than the hotel’s ($26/day, valet parking only) garage. See the Downtown San Jose Parking web page for details.
Parking lots particularly convenient to the Fairmont include the San Fernando & South Second Street Lot (30 E. San Fernando St.), which is an open surface parking lot suitable for taller vehicles. Just south of there is the Pavilion Garage (150 S. Second Street), which is a parking garage (max clearance 6’6″) and is very convenient to the hotel, as the exit stairs on First Street are right across from the Fairmont’s First Street doors. Both of these lots are maximum $20 day/$5 nights and weekends. (The parking garage attendant at the Pavilion said that July 4 counts as a weekend for this purpose; we have not checked with other lots.) Both of these lots list operating hours; however, we’ve parked overnight in both of them in the past, so you won’t be towed away for parking overnight, although you may not be able to exit during the wee hours of the morning. (The attendant at the Pavilion said that they’re actually open 24 hours despite what it may say on the web site.)
Another alternative, if you don’t mind not having convenient access to your vehicle, is to drop off your things (and your traveling companions) at the Fairmont and then driving to a nearby VTA Light Rail station with free long-term parking. The nearest such station is Tamien Station. You can park there for up to 7 days in the “Airport” spaces and ride light rail back to the hotel, which is located at the Paseo de San Antonio stop. Light rail fare is $2 for a one-way trip.