Located on the north side of St. Louis, Benton Barracks was one of the most important Union Army training camps in Missouri during the Civil War. The 150-acre complex was established in 1861, and contained barracks, warehouses, and numerous other buildings. A number of Missouri Union regiments were organized there, including some of the state’s African-American units. Soon after the war, the camp was closed and nothing remains of the wartime home of thousands of Federal soldiers.
The collection of troops and war material in the neighborhood of St. Louis in August  was of the most formidable character. At Camp Benton, immediately west of the Fair Grounds, extensive preparations were being made for the accommodation of a large body of troops, but only two regiments had taken post in this vicinity on the 21st.
-History of Saint Louis City and County: From the Earliest Periods to the Present Day
If we're looking for specific reasons why the war stunted the growth of baseball in St. Louis and across the nation, we find in this information something that I don't think has ever really been considered. Peter Morris, in But Didn't We Have Fun?, talks about how the availability of open field space in antebellum America was a significant contributing factor in the spread of baseball in the late 1850s. Here, early in the Civil War, we find that availability being curtailed. The land that would have been used to play baseball was being used as camp grounds by the Union army. In the specific case of St. Louis, we find the Union army using some of the very best and most popular pieces of land.
We see the impact of this in the contemporary source material. You don't see Lafayette Park and the Fairgrounds being used as baseball grounds after the summer of 1861. We now know that they were not available for use as baseball grounds because they were being used by the Union army. By August of 1861, baseball clubs were severely limited in their choice of grounds. The only decent grounds left was Gamble Lawn. There were other grounds being used in the city, such as Carr Park, the Laclede Grounds, and the Veto Grounds, but it appears that the best clubs, that had played at Lafayette Park and the Fairgrounds in the past, were pretty much limited to Gamble Lawn if they wanted to bring in a crowd. I would argue that this limitation in choice of grounds had a depressing effect on the growth of the game in St. Louis and I would imagine that we would see the same thing across the North in 1861.