Fangraphs has a discussion of the financial impact of Adam Wainwright being out for the entire season. The article estimates the impact as approximately $16 million based on lower attendance and a lower probability of making the postseason. The calculation suggests that Wainwright is worth about 4 wins and that would translate to the average team lower attendance worth about $10 million.
One problem with this analysis as I see it is that the Cardinals are not an average team. I have a paper that looks at the relationship between attendance and winning percentages but does not assume that all teams' fans are the same. In looking at the 12 National League teams that have been in existence since 1969, I found that the Cardinals' attendance was the least responsive to in-season changes in team performance.* In case you are curious, the Expos' and Dodgers' fan bases were the most responsive. My findings would suggest that the financial impact on the Cardinals will be less than it would be on the average team from which the numbers are derived.
Of course that does not mean that there is no impact, especially given that the Cardinals win expectation before the injury was right on the cusp of being a playoff team.
*The results for the Cubs, Astros and Giants are less reliable because of the number of sellouts those teams experienced. There are methods to deal with the sellout issue, but I was using an econometric technique to deal with a different concern and not compatible with the solutions to sellouts. It is possible that all three of those teams, especially the Cubs, would have smaller responses to winning than the Cardinals, but the Cardinals would still be near the top regardless.