Rotting wood covers their decks, their masts are flaked with rust, and their hulls are corroding.

New York’s last tall ships — once-proud symbols of the Big Apple’s rise to greatness — are in a shameful state of disrepair as the museum that’s supposed to care for them sinks in a Bermuda Triangle of debt and bad management.

Seaport Museum New York has closed its landside galleries and is looking to send its working ships to ports elsewhere for long-term storage.

The museum’s stationary ships — Peking, one of the biggest sailing ships ever built, Wavertree, a three-masted cargo ship, and Ambrose, a lightship that a century ago guided sailors into New York harbor — face an unknown fate.

“Those ships, which are emblematic of our heritage on the waterfront, are almost being left to rot,” said Roland Lewis, president of the Metropolitan Waterfront Alliance, a coalition of nonprofit groups.

Ambrose, built in 1908, is closed because of a leak below its waterline. Wavertree, built in 1885, is also closed and badly in need of repair.

On a sunny day last week, the 1911 four-masted Peking was the only vessel open to the public. It was often empty of visitors, despite the tourists thronging South Street Seaport.

Those who paid $10 admission saw plywood covering much of her deck, rusting iron masts and water-damaged paneling in the once-elegant captain’s salon. The fo’c’sle was closed off by metal sidewalk barriers, and the bowsprit — which points like a lance from the bow toward FDR Drive — is in danger of collapse.

A yellowing historic display referred to the early 1900s as “this century.”

Seaport Museum’s sinking fortunes are a bitter disappointment to the harbor’s old salts, who note that port museums in San Francisco, San Diego and Mystic, Conn., are thriving.

The museum was drowning in $1.8 million of debt at the end of 2009, its latest IRS filings show. Insiders believe the debt has ballooned since. Half the museum’s staff were let go in February.

Peter Stanford, who founded the museum in 1967, is rounding up a crew of volunteers and donors to save the ships — but says the museum’s current leadership isn’t interested in the help.

“They’re writing their own epitaph, and it’s a sad one,” Stanford said.

He believes the museum board and Mary Pelzer, its $124,000-per-year president, are more interested in internal politics and hobnobbing with the city’s elite than in the rugged work of recruiting donors and volunteers to keep its facilities shipshape.

“They’re interested in power, influence and who’s controlling. That’s a death rattle for a cultural institution,” Stanford said.

The museum declined comment, except to say it is “exploring various options” to maintain its vessels.