Vice President Bush has company on the airwaves in South Dakota, which holds its presidential primary Feb. 23, the same day as New Hampshire: Sen. Robert J. Dole (R-Kan.), Bush's closest rival for the 1988 Republican presidential nomination. Dole began airing a 30-second television commercial Thursday. He is depicted in soft hues in his first television ad that stresses his character and rural upbringing. Flashes of cornfields, uniformed veterans and a rising sun are blended with film of Dole's announcement in his hometown of Russell, Kan. "Growing up here is the greatest lesson you can ever learn," Dole says. "It's a lesson in character, stamina and concern for the needs of others. It's a lesson you take with you all your life."

The television ad is accompanied by a biographical radio spot that sets up a contrast with Bush's privileged background, asking, "What's the difference between the Republican candidates for president?"

The radio ad cites Dole's upbringing during the Depression, his two Bronze Stars for heroism in combat and the three years he spent recovering from World War II wounds that left him with a disabled right hand. It also says he led the fight to save Social Security and pass the 1985 farm bill. "Bob Dole for president. One of us," the radio ad concludes.

Don Ringe, media consultant to the Dole campaign, said yesterday that the ads are meant to reinforce "shared values" with South Dakotans.

The campaign ran radio ads featuring President Reagan praising Dole this week in New Hampshire. However, no ads are running in Iowa. Campaign officials said they are relying on a highly motivated organization in Iowa. "We don't really need to be on the air there right now," said adviser Bill Lacy.