It came as no surprise two years ago in an earnings call that Equity Residential, the giant apartment landlord, was bullish on Brooklyn. The borough “has come on strong with double-digit base rent growth,” said David Santee, chief operating officer of the real-estate investment trust.

Today, though, Chicago-based Equity is striking a different chord: While its Manhattan assets are generating strong demand and, in turn, higher rents, Brooklyn “is really dragging the portfolio down,” Mr. Santee said in a discussion of second-quarter...