As it says on the tin - how can you strengthen the U.S. Trade Union movement in a way that continues to the present day. While it's unlikely that Unions can achieve the dominance they do in some European countries, a solid goal would be having 30-40% worker enrollment in Unions, with no right to work laws on the books.
How can this be achieved? No Taft Hartley Act might be a good starting place.
We need to go back before Taft-Hartley, in my opinion, and avoid the First Red Scare. If you'll permit me an analogy, Taft-Hartley is Waterloo. By the time that Taft-Hartley is a consensus position--so uncontroversial it was passed over Truman's veto--the labor movement is already in a position of fatal weakness. If we snap our fingers and say the vote goes a different way, then the Act is brought up again in the next session of Congress, and it passes then.
The question is, then, how does left wing activism and the labor movement find itself in such a position of weakness to begin with? I would point to the First Red Scare as a grievous wound. About 600 activists were deported, thousands were arrested, Eugene Debs was arrested, Socialists were expelled from the New York Assembly. It totally destroyed the IWW. The strength of radicals within the movement never recovered, and consequently, when the Second Red Scare rolled around thirty years later, radicals were expelled from the Labor movement altogether, with the renunciation of Communism by union leaders required by Taft-Hartley and the expulsion of communists from within the CIO.
So, how do we avoid the First Red Scare? I don't think there's any way to avoid the Espionage or Sedition Acts, but those weren't strictly Red Scare laws, they were just laws that were used to great effect during the Red Scare. In my mind, there were three events that really whipped up hysteria: the 1919 anarchist mail bombing campaign, the Seattle General Strike, and the Boston Police Strike. The mail bombing campaign is an obvious cause for alarm. In the minds of those who were inclined to think of anarchists as dangerous people, it provided all the proof they needed to support their suspicions. The Seattle Strike was inspired by the successes of the Bolsheviks. While the strikers had a lot of reasons to strike, a communist influenced strike in the wake of a violent revolution in Russia,
while American soldiers were in Russia killing communists, was a very impolitic decision on the part of the unions that organized it. It made foreign communists--and not the labor movement--seem like a powerful force in American society. A force that was obviously subversive, destructive, and evil. If that wasn't enough, the Police Strike called into question the social contract. It made mass lawlessness and chaos a vivid threat in the minds of millions of Americans.
Clearly, people felt that things were spinning out of control, and A. Mitchell Palmer was just the man to grab the wheel and correct course.
So first things first. The Seattle Strike began as a shipyard strike, and the shipyard owners were initially willing to negotiate. Then a man named Charles Piez, who was the head of the local War Shipping board, threatened to cancel his contracts with the shipyards if any raises were granted (the guy was a major league asshole). He accidentally sent his threat to the unions instead of the bosses, and hell ensued. Best way to avoid all of this is to have him keep his mouth shut. The strike is resolved quickly and amicably.
Then we have to butterfly away the bombing campaign. It was separated into two parts: the April bombing campaign, which was at least partly inspired by the reaction to the strike; and the June bombing campaign, which was far more serious. Both campaigns were enacted by followers of an anarchist named Galleani, who was never directly implicated in the attacks, but may have had some knowledge. Even though Galleani was himself an advocate of violence, its plausible enough that he could make it known that he did not think that 1919 was the right time to take violent action.
Avoiding the Police Strike is a lot trickier. The Boston Police were subject to abysmal working conditions, and the police commissioner who would have been inclined to alleviate them, Stephen O'Meara, died in December of 1918 after months of health problems. The strike took place in September of 1919, so that's a long time to posit a longer life. The commissioner who replaced him, Edward Curtis, was a reactionary asshole, and he refused to negotiate or even recognize the right of the police to bargain collectively. So the best bet is to imagine someone else getting the job, someone who is inclined to be more flexible. Who that might be is anyone's guess.
On the flipside, Calvin Cooldige, who was the Governor of Massachusetts at the time, rose to national prominence because of his handling of the Strike. Butterfly the strike and you might butterfly away his presidency.
No Red Scare, no deportation of hundreds (including such luminaries as Emma Goldman) no arrests of thousands of activists, the IWW might be strong enough to force the AFL to deal with them (and thus avoid the crack up in 1924). Debs probably get released from prison sooner, and may very well live longer.
I think that's a very good start to giving the US a strong labor movement.