As Dropbox Inc. begins trading as a public company, it marks the first IPO by a Y Combinator company and a milestone for the accelerator behind other hit companies like Airbnb and Reddit. Although Y Combinator still nurtures scores of startups like those stars, it has busied itself with a wave of growth initiatives like raising capital from endowments, pushing harder into later-stage investing and backing more enterprise companies than ever before.

But in tackling those changes, Y Combinator walks a fine line to keep offering founders the level of personal attention that companies like Dropbox and Airbnb received in the program.

“I view it as my responsibility to not lose that spirit,” said Y Combinator President Sam Altman.

After launching in 2005, the program’s early batches had a couple dozen companies. At its demo day earlier this week, the program paraded 141 companies before investors crowded into a room at the Computer History Museum, as venture capitalists sat on the floor because the seats were all taken.

Once focused primarily on young founders of consumer apps, the accelerator program is now filled with startups that are more mature than their predecessors, and many of them come with different objectives in mind.

Two years ago, Y Combinator raised a $1.25 billion growth fund to invest in alumni coming out of its early-stage program. Y Combinator declined to comment on how much it had under management, but it has funding from institutions such as university endowments and a pension fund. Partners today collect management fees, in contrast to the early years, when the operation was funded personally by Y Combinator’s founders. At one point, they even used proceeds from the sale of some of Y Combinator’s stock in Dropbox, according to a person familiar with the matter.

In expanding beyond its early-stage program, the accelerator says its growth-stage program will function as a kind of “grad school” for YC companies that are post-Series A and have 50 to 100 employees.

Longtime followers of Y Combinator say the group has maintained its premier status by staying selective. More than 7,000 companies applied for the 141 slots in Y Combinator’s most recent class.

“Y Combinator is the Stanford or Harvard of accelerators,” said SV Angel founder Ron Conway.

Fifteen Y Combinator companies have gone on to raise funding or be acquired at a valuation of $1 billion or more. It remains to be seen whether these highflying alumni will have exits that live up to their lofty private-market valuations.

Dropbox, which commanded a $10 billion valuation in its last funding round, has set an IPO target price range that implies a market capitalization approaching $9 billion.

Software startup Zenefits Inc. was, for a time, another Y Combinator-backed hit. Once valued at $4.5 billion, Zenefits later slashed its valuation to $2 billion in 2016 in the aftermath of scrutiny by regulators. Airbnb, which at $31 billion has the highest valuation of any Y Combinator company, has said its long-awaited IPO won’t happen until 2019 at the earliest.

The most recent batch in Y Combinator’s accelerator had more B2B than consumer companies, with 17% in biotech or health care, including Nectome, a company that aims to back up customers’ brains.

Harj Taggar, a Y Combinator alumnus with two different startups, said the camaraderie and support he felt from the other entrepreneurs and founders defined the program in its early years. He said he was close to the founders in 2008, and several of them lived in the same San Francisco apartment tower, which they dubbed the “Y Scraper.”

Years later in his second stint, Mr. Taggar said, the program was much larger and not as tightly knit, but that the exponentially larger alumni network helped scale up his company. Some of his first customers included such Y Combinator alumni as Dropbox, Flexport and Reddit.

“It felt like going through a machine,” Mr. Taggar said, noting the program in his second stint was much more structured than the first time.

Many sector-specific accelerators that specialize in areas like health care. Other well-known accelerator programs, such as Techstars, often take a more thematic approach.

But Y Combinator Chief Executive Michael Seibel said his goal is to help founders with everything from running a fundraising process to the stress of running a startup.

“You don’t go to YC because they know more about your core product, core service, core customer than you do,” Mr. Seibel said. “You go to YC for everything else.”