I have always considered this to be a bit of a historical "what if" question, but historians generally seem to agree that reconstruction would have been quite different with Lincoln at the helm.
Kearns Goodwin backs up the claim in question by selecting a few quotes from Lincoln just a few days before his death. It is evident that Lincoln's intentions in reconstruction are to restore and rebuild the country without animosity and with a kind hand. In a meeting between himself and Edwin Stanton on the subject of reconstruction, Stanton recalls:
"There were men in Congress who, if their motives were good, were nevertheless impracticable, and who possessed feelings of hate and vindictiveness in which he did not sympathize and could not participate. He hoped there would be no persecution, no bloody work after the war was over."
Lincoln also insisted that they not punish the Rebel leadership or resort to hangings.
"Enough lives have been sacrificed. We must extinguish our resentments if we expect harmony and union."
Doris Kearns Goodwin, Team of Rivals (New York: Simon & Schuster, 2005), 732.
The assumption is that his popularity following the war and his sway over the Republican Party would have followed his tone of amicability. It's hard to know. Lincoln could have undergone the same treatment as Andrew Johnson, his successor, who certainly put up his fair share of fighting against the radicals and ended up being impeached for it.
However, as I explained in a thread earlier this week, Andrew Johnson, his Vice President, was in fact a Democrat. As you can tell, it gets complicated.
A side note, while Booth was on the run after the assassination, one of his conspirators was able to gather a few newspapers from the South so that Booth could read the responses to his deed. Southern editors could have realized that there would be a backlash in the reconstruction process or genuinely felt this was a needless death of a respected leader, but either way, the South's condemnation of Booth's actions was incredibly swift and unanimous. Booth was astonished and failed to understand why he was being condemned "for doing what Brutus had done." Booth may have been seen Lincoln as a tyrant during the war, but he failed to understand what his fairness in reconstruction could have meant to the South.
If you're looking for more Booth-Brutus talk:
I haven't really read much on the reconstruction politics outside of Kearns Goodwin, so please add them on.
We are firmly in the realm of speculation here, but I don't agree with Goodwin. Does she really think that in the face of repeated instances of Southern intransigence (and here I refer to everything from sending Confederates right back to Congress, to attacking Freedmen's Bureau agents, to extralegal violence against blacks, and the Black Codes), Lincoln would not have gone along with Congress in making the terms of Reconstruction harsher?
I find that hard to believe. Nothing in his career as President indicates a willingness to surrender the authority of the government to illegal violence.