Certainly as president, Lincoln had ample opportunity to see a baseball game. Before, during, and after the war baseball clubs competed on the President's Grounds near the White House in Washington, D.C...According to [Winfield Scott Larner,] Lincoln and his son Tad watched the contest from a spot along the first base line, cheering with their fellow fans and also receiving an ovation from the crowd.
At about six o'clock, the President, who was prevented from appearing earlier on account of the semi-weekly Cabinet meeting, came on the ground and remained until the close of the game (Washington 28 vs Brooklyn Excelsiors 33), an apparently interested spectator of the exciting contest.
I'm inclined to give some credence to this story for a few reasons. First, Lincoln was a man who enjoyed athletics and athletic contests. He had a reputation as a good athlete and there are numerous accounts, told by people who knew him, of Lincoln participating in athletic contests, including town ball (which I'll cover in another post). He was also a man who enjoyed people. He enjoyed sitting around with people. He loved talking to people, telling and listening to stories. I believe that it would be in Lincoln's character to go to a baseball game and to enjoy himself while there.
Second, Lincoln certainly had the opportunity to go to a game. As Kirsch noted, there was a ballpark in his backyard. Also, while Lincoln was obviously under a great deal of pressure during his time in office and the crush of business was, at times, overwhelming, there were also periods when there was nothing going on. There were moments during the war when the armies were inactive and when Congress wasn't in session. During these times, Lincoln really had rather little to do. While I doubt that Lincoln went to a baseball game during the first few days of July in 1863 or when Republicans in Congress were trying to force various cabinet members from office or during one of the diplomatic crises with England, the Lincoln administration was not one crisis after another, twenty-four hours a day and seven days a week. There were quiet moments when Lincoln was able to find time for rest and relaxation and I don't think it's out of the realm of possibility that he could have taken his son to a baseball game.
Lastly, and most importantly, there was Lincoln's relationship with his son Tad. Lincoln's love of children is well documented and, while his relationship with his oldest son Robert was rather strained, he was particularly found of his younger sons, Willie and Tad. A doating father, his younger sons were spoiled by Lincoln and his wife. Donald, in his biography of Lincoln, writes about the relationship between Lincoln and his son:
Lincoln drew much comfort from Tad, to whom he became even more attached after the death of Willie. He spent much time playing with the boy, and he helped him raise his kitten and train his dog...Because of his speech defect most people could not understand Tad, but his father always could-and he knew how frustrated the child became when he could not express himself...In turn, Tad adored his father, and he would often hang around the President's office until late at night, sometimes falling asleep on one of the couches or chairs. When Lincoln got ready to retire, he would pick the boy up and carry him off to his big bed, where Tad now mostly slept.
However, it must be stated that there is no primary source evidence that Lincoln attended a baseball game in Washington while he was President. As I've argued, the accounts of Lincoln attending a game are reasonably believable and shouldn't be dismissed out of hand but all we can do is speculate. Personally, I'd like to know more about the baseball grounds in Washington, about the history of the game in Washington during the war years, and find a date for the Washington/Brooklyn game. I think that adding that information to what we already know about Lincoln would strengthen the argument that he attended a baseball game while President.
One more thing that I'll just throw out there without any evidence to support it and without really thinking it through: Lincoln visited the Army of the Potomac on numerous occasions, often staying overnight at the camp. Given all we now know about baseball during the Civil War and about specific instances of baseball activity among the troops, isn't it possible that Lincoln may have seen a baseball game being played during one of his visits? I have no idea if he did or did not or if the scenario is entirely realistic but the thought occurred to me and I figured I'd throw it out there.