The tourists arrived for the three-Test series amid a storm or controversy following the omission of paceman Andre Nel to fulfil the team's quota of non-white players.
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Nel was furious at being dropped, prompting speculation he will quit the international game.
Fellow paceman Charl Langeveldt, a mixed-race player chosen in Nel's place, backed out of the tour as a result of the tension created by his selection.
Black fast bowler Monde Zondeki was named as replacement. The first Test begins in Chennai on Wednesday.
"It is a disappointment, [but] we talked to them," Smith told a news conference when asked about the selection saga.
"But we are a mature team and we'll overcome the challenge."
Smith said South Africa are looking forward to a close series in India with his team hoping to carry forward the momentum gained from successes in Pakistan last year and Bangladesh recently.
India put up a solid display on their Australia tour, losing an acrimonious Test series 2-1 before winning the one-day tri-series, where they beat world champions Australia 2-0 in the finals.
"It is going to be a big challenge," Smith said. "I know India have come back after a great tour of Australia."
However, he was confident there would be no verbal clashes like the ones that marred India's tour.
"It will be a battle of cricket and not a battle of nerves," he said.
Former South Africa opening batsman Gary Kirsten will begin his term as India coach. The tourists are upset that he has lured mental conditioning coach Paddy Upton.
Coach Mickey Arthur said Upton's presence in the Indian camp could be a disadvantage because he knows the South African players well.
Ahmedabad and Kanpur will stage the final two Tests.