I had to look up where that excerpt came from to get a clearer understanding of the meaning of sensitively in the context you mentioned. I believe the next paragraph explains the context of the word:
"Training powerful tools and techniques on a work of art can fetishize it, treating it as a relic whose every mark takes on theological import. But it can also humanize the object by making it seem contingent -- not an unalterable masterpiece but a resting point along a journey. In the exhibition, which includes a room dedicated to the technical analysis of the paintings, the curators, Ron Spronk and Harry Cooper, have laid bare the hesitations and rethinkings that lie behind these serenely abstract works."
In this case, I think sensitively in the article refers to handling the art and applying the new technologies in a way that promotes care, perceptiveness, and appreciation. How the technology is applied is properly considered. It is important to handle the art with care, and take the piece as a whole rather than the components only discoverable with new technology. It is stressed that individuals should not have an excessive obsession about a particular aspect of the art based on new technological findings and instead remember that old-fashioned scholarship and curatorial judgement is indispensable.
"In this case technical analysis revealed something unexpected in the character of the notably reserved Mondrian. ''What really surprised me,'' Mr. Spronk recalled, ''was the vigor and the incredible energy, how laborious these reworkings were and to what length Mondrian went to attain these paintings. He literally attacked them.'' But while acknowledging the insights gained by peering beneath the skin, Mr. Cooper strikes a cautionary note. ''There is the fallacy with this kind of work that you can reach a kind of ground zero,'' he warned. ''You just can't.'' In other words, there is no high-tech substitute for old-fashioned scholarship and curatorial judgment."