A seven-minute walk from Nishi juhatchome (18 chome) Station on the Tozai Line (Exit 4).

JR Bus or Chuo Bus (Sapporo–Otaru/Teine route)
A four-minute walk from the Doritsu Kindai Bijutsukan bus stop

A 5-10 minute taxi ride from Sapporo Station. Tell the driver that the museum is  "on the north side of the garden of the Governor’s Official Residence, on the Kita 3-jo dori Avenue side."

There are two free parking spaces at the museum, one of which is reserved for persons with disabilities. 

Parking is also available at the Big Shine 88 Kita 1-jo Car Park (Kita 1-jo Nishi 15-chome) for a discounted fee. 
Price
(Until May 31, 2018) 120 yen per 20 minutes (120 yen for each additional 20 minutes)
(From June 1, 2018) 130 yen per 20 minutes (130 yen for each additional 20 minutes)

Discount for museum visitors

Stamp your parking ticket at the stamp machine near the west entrance (next to the public telephone) on the first floor of the Museum of Modern Art, or at the stamp machine at the reception desk for the exhibition rooms of the Migishi Kotaro Museum of Art, and you will be entitled to 20 minutes free parking from the time you enter the parking lot.

Discount for persons with disabilities

If you show your Physical Disability Certificate, Medical Rehabilitation Handbook, or Mental Disability Certificate at the counter of Big Shine before you pay for parking, you will be given one hour free parking from the time you enter the parking lot. 
Note that this discount cannot be used in combination with the discount for museum users mentioned above. (If you stamp your parking ticket with the stamp machine at the museum, you will not be able to apply for the discount for persons with disabilities.) 
Note that parking is prohibited on the streets and sidewalks in front of the museum.