First of all, I'm not a US labor historian; it's as far from my specialization as it gets. With that in mind, I'll provide my opinion, hoping someone else will contribute with a better theoretical underpinning and fill in some of the details.
I would argue for the 1981 Air Traffic Controller Strike as a turning point. PATCO (Professional Air Traffic Controllers Organization), the air traffic controllers' union, wanted to negotiate for better pay, shorter working hours and and opting out of the civil service clause (ie, not being able to go on strike). When PATCO announced an undeclared strike, then-president Ronald Reagan deemed it illegal and promptly fired 10,000+ workers represented by the union. The fired workers were then allowed to reapply for their jobs. Some were jailed. Prof. Joseph McCartin has argued that the firing of the air traffic control workers was the most monumental event in 20th century US labor history. The mass firing was a turning point in labor-management relations, leading to a long term decline of labor unions and a dismantling of collective bargaining rights that originated with the New Deal era Taft Hartley Act. In terms of legislation, it also gave a green light to a spate of "right to work" laws that diminished, if not outright destroyed the rights of state employees to negotiate with their employers. More broadly, I would also argue that it was the first, and possibly the most successful blow by the political right against New Deal legislation, so the symbolic importance of the failed PATCO strike cannot be understated.
Joseph McCartin, Collision Course: Ronald Reagan, the Air Traffic Controllers, and the Strike That Changed America (Oxford, UK, Oxford University Press, 2011)
Could you specify what anti-union sentiment you're talking about?
I went to a talk recently by J D Chandler and J B Fisher, authors of Portland on the Take. Their story starts with labor problems in the 1930's, specifically the dockworkers strike that shut down Pacific coast ports for a couple of months. The strikers won, and the many well-organized workers with communist sentiments were a large reason why. The reaction to this by the malefactors of great wealth, who already had anti-communist red squads operating in cooperation with police departments around the country, was a campaign of infiltration of the unions by informers and agent-provocateurs, and a willingness by law enforcement to allow gangsters to take over unions in return for cutting the leftists out and paying graft to police and municipal officials. Thus positioned in the middle of an extremely corrupt power structure collecting government salaries, payoffs, extortion, taxes, local monopoly profits (particularly from vice), union dues and other payroll deductions by force and coercion from a very large portion of the population, the unions were bound to have some image problems.
Sorry, that's probably a lot more than you wanted to know.