WEST COLUMBIA, S.C. — The problem began as soon as Bernie Sanders walked into the dining room of the revered and predominantly black Brookland Baptist Church here. Instead of flocking to him, as supporters do at his large college rallies, many of the church’s 780 members present looked up for a moment, then quietly went back to eating their Sunday feast — unmoved as Mr. Sanders, the senator from Vermont, tried to work the room.

Mr. Sanders delivered remarks at a microphone next to a buffet table offering chicken, collard greens and dinner rolls.

“We have, in America today, a broken criminal justice system,” Mr. Sanders said at the microphone, pausing briefly after this line from his stump speech, which is usually met with applause. Here it garnered very little, and the line for the food kept moving. Brookland Baptist Church proved a tough crowd.

His visit here underscored Mr. Sanders’s challenge in strengthening his support among black voters in South Carolina, where the majority of Democrats casting a ballot in the state’s primary on Saturday will be black and where his rival for the presidential nomination, Hillary Clinton, seems to be holding on to a sizable lead.