It sounds awful every time we talk about it. How dare the state cut off its annual $2 million handout to the Rose Fitzgerald Kennedy Greenway?

But Charlie Baker is right. It’s time.

The Greenway has overstayed its welcome in the state budget, like your adult kid who won’t leave the nest because the rent is free and the laundry always gets done.

Now, I’m not proposing that the state go cold turkey this summer, when its subsidy agreement expires. That would be a bit cruel and unusual, since the state kicks in about 40 percent of the downtown park’s budget. Instead, there should be an organized phase-down, but not a complete cut-off, since the state gets something from the nonprofit Rose Fitzgerald Kennedy Greenway Conservancy, which manages and maintains the state-owned, 1.5-mile park above the O’Neill Tunnel.

So who should make up the difference when the state steps back?

Baker had some thoughts on this in 2008, back when he was the CEO of Harvard Pilgrim Health Care and sat on the conservancy’s board.

“As the value of the land surrounding the Greenway increases, I would hope we would see more enthusiasm from the abutters in the future,’’ Baker told Commonwealth Magazine.

For those of you who don’t speak wonk, Baker is talking about all of the commercial property owners who touch the Greenway — and whose tenants­ benefit from the lush, organic grounds.

Talk to property owners and they will tell you that they give plenty. But I would argue not enough. The number is almost too embarrassing to print. Last year, they contributed $130,000 toward the Greenway’s $5 million operating­ budget. And that was a good year. Most other years — the conservancy was formed in 2004 — property owner funding totaled just $100,000.

Let’s put that in perspective.

There are billions of dollars of real estate wealth along the Greenway. These building owners already pay property taxes — which should go toward maintaining the Greenway — so that’s why they don’t feel obligated to contribute more.

But with the value of their properties soaring, in no small part due to the blossoming of the Greenway, it’s hard not to ask these landlords to give more, because ultimately it’s good for their bottom lines.

There’s also precedent to do more.

In Downtown Crossing, property owners in 2011 opted to pay higher taxes to fund a business improvement district to clean up the gritty neighborhood.

Owners in a 34-block zone pay a special tax of $1.10 for every $1,000 of assessed property value up to $70 million, and 50 cents on every $1,000 of value above that.

Last year, property owners in the business improvement district paid anywhere from about $300 to $250,000 each, generating about $7.2 million for Downtown Crossing that can be used on everything from landscaping to welcoming visitors to sidewalk improvements.

Not that long ago, Downtown Crossing was a mess, with a gaping hole in the heart of the city when a developer halted construction during the Great Recession.

Today, the neighborhood is one of the city’s hottest residential and commercial districts, bustling with restaurants and ritzy condos, and the business improvement district plays a pivotal role.

The conservancy wanted a business improvement district, too, and pushed for one five years ago, but property owners balked. It’s back on the table, but these are hard to form. Under state law, 60 percent of the property owners in a designated district — and who represent at least 51 percent of the assessed value within the district — must agree to the special tax.

Rosemarie Sansone, chief executive of the Downtown Boston Business Improvement District, reminds me it took three attempts before property owners agreed to form that one. Even so, it takes a campaign to get everyone’s buy-in.

“It’s really driven by what the commercial owners want,’’ she said.

Rick Dimino, CEO of A Better City, a business group that represents many Greenway abutters, wasn’t keen on a business improvement district, but he said owners are open to the idea of a more formal structure of payment.

“We have consistently been interested in looking at ways to be helpful beyond the current level of contribution,’’ he said.

But beyond business upping the ante, so can the conservancy. The group has done a terrific job in fund-raising since its inception — $28.5 million in philanthropy — as well as in generating additional revenue from sources such as food-truck leases and carousel ticket sales.

Most of all, the Greenway has really come into­ its own, with public art installations and other programming attracting about 1.4 million visitors last year.

But there’s more it can do. One area of growth: corporate sponsorships. If the South Boston convention center’s Lawn on D can get $250,000 from Citizens Bank, certainly the Greenway can ink a few deals like that.

Take a look at New York City’s Bryant Park, which gets support from Bank of America, Google, and Facebook. My favorite? Southwest Airlines, which sponsors the southwest corridor of the popular Midtown park. Sponsorship takes the form of underwriting a festival or seating and umbrellas with logos. While the park is public, it takes no government money.

Dan Biederman, the president of Bryant Park, offers this advice to the conservancy:

“Try to become self-sufficient as soon as possible. Take advantage of the good work you have done.’’

In other words, the conservancy has already done the hard part by remaking a no man’s land into the people’s park. Weaning the Greenway off state money might just light a fire under everyone to be more creative in funding its future.

Shirley Leung is a Globe columnist. She can be reached at shirley.leung@globe.com. Follow her on Twitter @leung.