The Jaffe-Quinn manifesto really had little to do with string theory, but a lot to do with topological quantum field theory, especially 3d tqft. I remember Frank Quinn talking about this at length during a hike at the 1991 Park City summer school. He was lecturing there on topological qft, see
"Lectures on Axiomatic Topological Quantum Field Theory" in "Geometry and Quantum Field Theory, IAS/Park City Mathematics Series, Volume 1", edited by Daniel Freed and Karen Uhlenbeck.
The sort of thing that was worrying Quinn was:
- Witten's great paper on "Supersymmetry and Morse Theory", which was published in a mathematics journal, the Journal of Differential Geometry.
- Witten's Fields medal winning work on the Jones polynomial and Chern-Simons theory.
Quinn explained that at the beginning of his career he had been heavily influenced by the work of Thurston and Sullivan, but found that trying to emulate them had led him to lose track of what he precisely understood and what he didn't, requiring a painful period of getting back to a more rigorous way of working. He was worried that losing the distinction between works like Witten's and truly rigorous work would lead others to the problematic situation he had found himself in as a young mathematician. In the end, I think Atiyah's response won the day: he argued that mathematicians were fully capable of protecting their virtue while interacting with physicists. Shortly after this exchange, those topologists in the math community who were skeptical about the importance of what Witten was bringing to mathematics were conclusively won over by the Seiberg-Witten equations.
But the example set by Quinn of how to do TQFT in the end has largely won out. There was an attempt to teach mathematicians the actual QFT behind Seiberg-Witten at the IAS in 96/97, but I don't think it was very successful. These days both TQFT and the Seiberg-Witten equations remain very important ideas in topology, but they're pursued with conventional standards of rigor. Mathematicians have gotten used to taking physicist's QFT arguments and extracting and generalizing those parts that can be made rigorous and fit into the evolving mathematical tradition.
As others have mentioned, for the case of string theory, mirror symmetry is probably the best example of an idea coming out of it that has had a huge influence in mathematics. Yau's recent popular book "The Shape of Inner Space" contains lots of other examples of the interaction of math and physics surrounding Calabi-Yau manifolds.