"I can't promise that there won't be another accident - how can I?" he told an inquest at Canterbury, Kent.
"But we will try to reduce the odds. We will keep the ethos and spirit of the place at the same time. We don't want to abandon that. We don't want another boring zoo where the keepers just become jailers and functionaries rather than companions."
The jury of seven men and three women returned an accidental death verdict on Mr Smith, 32, of Goodneston, near Canterbury. The inquest heard that Mr Smith would have died instantly from a broken neck after being bitten by a two-and-a-half-year-old Siberian tiger, Balkash, at Howletts Zoo, near Canterbury.
Mr Aspinall, 68, who gave his address as Howletts Zoo, was questioned by Roy Warne, counsel for Mr Smith's widow, Debbie, 22, about the risks faced by keepers going alone into the tigers' enclosures.
He said: "We live in a dangerous world. Everyone gambles with their lives all the time. Going in with tigers is not as risky as it seems." Mr Aspinall added: "If you go to a football match, you gamble."
He himself had been alone with tigers many times over the years, although now he was getting older and weaker he preferred to have his head keeper with him. He had 25 tigers at Howletts and nearly as many at his other zoo at Port Lympne near Hythe, Kent.
The death of Mr Smith was "a great disaster for us", but it was the first tragedy of its kind, he said. The tiger had killed two keepers in 1983 but had scrambled over high fences to reach them. Mr Aspinall said he had run Howletts for 37 years and Port Lympne for 22 years. "We are pioneers in the work of building friendship between the mammals and ourselves."