The raid at Harpers Ferry had several immediate effects which can be seen as not only causative with regard to the war, but also having a bearing on the way the war was fought and, possibly, its outcome.
Prior to Brown's raid, the Southern militia system barely existed. A man attempting to start a rising of enslaved people, though it was easily stopped, led the Southern states to expand and to organize their militias with large numbers of volunteers. These militias then became the nuclei of the Confederate armies when war broke out a year and a half later. The raid also convinced Southerners that not just John Brown but other Northerners intended to free the enslaved people and to abolish the institution of slavery, not merely to "contain" it by excluding it from the territories, as the platform of Lincoln and his party would be the following year (1860).
The reaction in the North to Brown's raid was that Brown began to be seen as a martyr even by people who were not especially committed to abolition. Though his tactics were decried by most people (including Lincoln), the cause for which Brown stood was central to the conflict between North and South. This had always been the case, but the spectacular (and doomed) act of violence perpetrated by Brown had the effect of inflaming people on both sides. All of these factors led, however indirectly, to secession first by the Southern states a year later and to a protracted war in which both sides would not back down over the issues at the root of the conflict. In answering your question as to whether the raid affected the outcome of the war, we cannot give a direct yes or no because there were so many intervening things that happened in the span from October, 1859 to April, 1865. Brown's legacy, throughout the conflict, was an inspiration to many of the Union troops, as the song "John Brown's Body," which then became the "Battle Hymn of the Republic," indicates. But that legacy also could have made the Confederates wish even more to fight to the death, given their fear that what Brown intended—the liberation of the enslaved people—was inevitable with Union victory, as it in fact was.