Like almost all empirical economics studies, we don't have the counter-factual and so we are forced to try and do the best we can with the data we have. In this case they use the newly fashionable regression discontinuity design. In essence, they look at businesses where unionization just barely passed, versus businesses where it just barely failed. Why, well unionization is not likely to be random and unions could form at more profitable businesses that are more likely to pay higher wages. So if you see higher wages at union shops it is impossible to know how much is due to unionization and how much is due to the general profitability of the firm.
What do they find?
The analysis finds small impacts on all outcomes that we examine [business survival, employment, output, productivity, and wages]; estimates for
wages are close to zero. The evidence suggests that—at least in recent
decades—the legal mandate that requires the employer to bargain with a certified
union has had little economic impact on employers, because unions have been
somewhat unsuccessful at securing significant wage gains.
Thus the answer is not much. Unions in this period didn't seem to be effective at bargaining up wages and thus don't have much of an impact on firm performance.
But there remains an open question: in cases where unions did seem to win wage concessions was this at the expense of firm performance? I'll try and find evidence next time...