Cartels and the Secret Ballot
First, the act strengthens unions, and unions are a cartel. When cartels exert their monopoly power, they raise wages above the equilibrium level, leading to a variety of inefficiencies and inequities. As Larry Summers once put it, "Another cause of long-term unemployment is unionization. High union wages that exceed the competitive market rate are likely to cause job losses in the unionized sector of the economy. Also, those who lose high-wage union jobs are often reluctant to accept alternative low-wage employment."
Second, the act would allow unions to organize without a secret ballot, as long as a sufficient number of workers put their names on cards endorsing a union. Union organizers would be able to use strong-arm tactics to get workers to say they support a union, even when privately the workers don't. In light of the sometimes violent history of the labor movement, this concern seems very real. As a matter of procedural fairness, I cannot understand why one would oppose a secret ballot.
My guess is that if you polled economists, you would find more opposed than in favor of the proposed legislation. But that is just a guess, as I have not seen such a poll conducted.