And they stressed the storm will keep causing problems as it spreads east through Friday.
"This is a dangerous storm," Gov. Mike Rounds told reporters in a telephone conference call early Thursday evening.
"Western South Dakota is basically under a no-travel advisory."
The storm already has dropped 45.7 inches of snow near Deadwood, in the northern Black Hills. Reports of 10 inches to 2 feet of snow were received from many West River counties. In some towns, residents reported drifts were blocking their doorways.
In Shannon County, in the southwestern corner of the state, 20-foot snowdrifts were reported on the Pine Ridge Indian Reservation.
The snow fell at a rate of 3 inches an hour early Thursday, the weather service said, and downtown Rapid City reported a 78-mph wind gust late Wednesday.
Dozens of schools, agencies, businesses and attractions were closed in the region - including Mount Rushmore National Memorial.
Rounds said most of the dozens of vehicles stranded along Interstate 90 have not been moved. Some have been stranded for more than 24 hours, he said, adding that search teams can't get to them because of zero visibility.
"We cannot see a thing in many areas where we're out actually searching for people," said Tom Dravland, state Public Safety secretary, who added that top speed for some rescue crews was as little as one-half mile per hour.
Dravland said he did not know how many people are stranded. The Highway Patrol has responded to more than 400 calls for assistance, including 10 crashes.
No fatalities had been reported by late Thursday afternoon.
Interstate 90 is closed from Mitchell west to 50 miles past the Wyoming state line.
Many of the stranded motorists have cell phones, but Rounds said some cell phone towers in the affected area were running on battery power because electricity has been lost - and that some of the towers' batteries were running low.
Officials said the problem is heavy snow and wind caused many power outages. But repair crews can't get to the downed lines because of the blizzard, Rounds said. Some people will be without power for several days, he said.
None of the state officials who talked with reporters Thursday evening had an estimate of the number of people without power.
According to the South Dakota Rural Electric Association, eight cooperatives reported outages and damage to their systems. The six with the most outages were Butte Electric Cooperative of Newell, Grand Electric Cooperative of Bison, Lacreek Electric Cooperative of Martin, Moreau-Grand Electric Cooperative of Timber Lake, West Central Electric Cooperative of Murdo, and West River Electric Association of Wall.
Co-ops in Custer and Mission reported fewer outages.
Reports Thursday indicated at least three co-ops would need help from others to repair the damages: Grand Electric, Lacreek Electric and Moreau-Grand Electric.
Wet, heavy snow that clung to roads, trees and power lines was the source of the trouble, the South Dakota Rural Electric Association said.
Black Hills Power reported that about 7,000 of its customers were without electrical power Thursday, most of them in the Rapid City, Black Hawk and Piedmont areas. Some phone service was out as well.
Butte Electric officials reported broken poles and downed wires in Newell and Butte County.
The whole weather system should wind down Friday, said Greg Harmon of the Sioux Falls National Weather Service office. He said the winds should subside in the west early Friday and in the east later in the day. Low temperatures were forecast for the low to mid 20s statewide overnight.
The snow started Wednesday afternoon and was moving east. Winter weather advisories were in effect for the western edge of the state and the southeast corner Thursday night. Most of the rest of South Dakota was under a blizzard warning.