At the height of the summer tourist season, as many as 28,000 people walk through the Sistine Chapel and the rooms in the Apostolic Palace decorated with frescoes by Raphael in the course of a day. In the Sistine Chapel alone, dust left by visitors over the course of a year—much of it the accumulation of microscopic bits of human hair and skin—takes weeks to remove in an annual cleaning, Ms. Jatta said.

To minimize the damage such crowds inflict on the works of art and improve the visitors’ experience, the museums have installed new energy-saving LED lighting in the Raphael rooms and will soon install new air conditioning systems there, following a similar light-and-air project undertaken in 2014 in the Sistine Chapel.

Working with tour guides and the companies responsible for a large proportion of the visitors, Ms. Jatta hopes to spread out the flow to make the experience in those areas less stressful for people and the artworks themselves.

That means getting people to spend more time in lesser-known parts of the museum complex, including the Etruscan Museum, one of the most valuable collections of artifacts from that ancient Italian civilization. A large elevator will be installed to facilitate access to that museum, which in addition to the glories of its collection, boasts a 360-degree panorama of Rome.