Jesse Jackson ran a much more inclusive primary campaign in 1988 based on his Rainbow Coalition and was not perceived as solely 'the black candidate.'
Jackson Share of Votes By Whites Triples in '88
https://www.nytimes.com/1988/06/13/us/jackson-share-of-votes-by-whites-triples-in-88.htmlWhile 2/3 of the votes he received were from blacks, he still received over 2 million non black primary votes in 1988. (Michael Dukakis received 9.7 million votes.)
https://www.thedailybeast.com/the-unsung-heroism-of-jesse-jackson“Economic common ground” became a refrain of the Jackson campaign, long before the “99 percent” became part of the public vernacular. “We said, ‘if we could leave the racial battleground for economic common ground, we can find the moral higher ground.’ That’s how we got the rhythm.”
The wide range of Jackson’s clarion call brought together former enemies in American life. Jackson recalled meeting a group of white supporters in Alabama who said, “We were with you in Selma.” After Jackson thanked them, they said, “You don’t understand, we were on the other side.”