Mr. Del Vecchio credits Brooks Brothers’ 27 airport shops, operated by a licensee, for helping win back businesspeople who had rejected Brooks Brothers in the 1990s. He calls the shops a “great showcase” for the brand. (In the 2009 movie “Up in the Air,” George Clooney’s traveling businessman character lingers over a display of striped ties at a Brooks Brothers airport shop.)

Brooks Brothers has also reached out to established fashion designers for exclusive, high-profile capsule collections — Thom Browne from 2007 to 2014; Zac Posen for women’s wear since 2016 — but its business remains rooted in its classic men’s wear, which accounts for 80 percent of its business.

Dress shirts, now in about 1,000 varieties, have long been the calling card of Brooks Brothers, accounting for 30 percent of its sales. In a nod to contemporary trends and to buffed, young guys, the shirts come in four fits: the Traditional, the Madison, the Regent and, the slimmest, the Milano. (Mr. Browne, famous for his tightfitting men’s suits, helped steer Brooks Brothers toward slimmer silhouettes, said Lou Amendola, the store’s chief merchandising officer. “Today over 50 percent of our business is now in slim shirts and slim suits,” he said.)

Charles Moore, founder and president of the Banc Funds, a private equity firm in Chicago, said he had stopped wearing Brooks Brothers dress shirts for several years because “the quality of the shirt fabric suffered and the collar wasn’t fitting.” He shifted to $200 custom shirts until a few years ago when he returned to Brooks Brothers, for its trim Regent silhouette, which was new to him.

“I like the fine Supima cotton and the way the shirts ride on your neck — the spread collar and the button-down collar,” he said. For around $80, “they’re great value for the money.”

‘We Are Authentic’

The privately held Brooks Brothers has posted profits for 13 of the last 17 years. For the past three years, annual sales have hovered around $1 billion, with profits at a break-even level, according to figures provided by Mr. Del Vecchio. (In the current challenging retail market — with Ralph Lauren Corp. and Abercrombie & Fitch closing down stores, and J. Crew getting rid of its entire top management team to try to reverse that company’s revenue slide — steady results can be considered something of an achievement for Brooks Brothers management.)