Because the song is based on an actual event, many Dylan sources erroneously report that that's how things actually went down. It isn't, though. The swelling of people didn't cause a sinking - it caused a stampede/riot.
Before boarding the Hudson Belle, the attendees started figuring out that many of them held fake tickets, which would mean many of them wouldn't be able to actually go on the trip. So, when the Hudson Belle pulled up to the pier, people started pushing and shoving to get onboard. Upwards of 150 police officers arrived to try to contain the situation and prevent people from boarding.
What followed was described as a panic and a riot, but that seems to be using the terms generously. In the surge to get on the ship, many people did get trampled, but the most serious injury was a broken leg by 26-year-old Coldridge Barbour
of 128 West 115th Street (back in the day newspapers printed people's full addresses). Barbour was a musician, a member of a Trinidad steelband that was supposed to play for the attendants.
Other than Barbour, 75 people were treated and released on-site. Six went to Knickerbocker Hospital
but were quickly released. The Knickerbocker was the location of a Cinemax series that was titled The Knick
, which ran from 2014 to 2015.
The ship's captain decided to cancel the trip to Bear Mountain. All those who paid for legitimate tickets were refunded, but those with fraudulent tickets were out of luck. According to The Daily News
(June 19, 1961), they'd paid $3 for each adult ticket and $1.50 for kid's tickets. In today's dollars, that's about $26 or $13, respectively.
Noel Paul Stookey (the "Paul" of folk trio Peter, Paul, and Mary) read the story in a June 19 paper. He brought it to Dylan, and by the next day, Dylan had written "Talkin' Bear Mountain Picnic Massacre Blues."