You can check out this article which talks about both sides of the Cleveland affair scandal. Cleveland's side of the story was one of merely being a helpful friend:
At the time, the campaign provided this rationale: Cleveland was a bachelor, and Halpin had been rather free with her affections, including with some of Cleveland’s friends—prominent Buffalo businessmen all. As the only unmarried man of the bunch, Cleveland, though not certain the child was his, claimed paternity and helped Halpin name the boy and place him with a caring family. Really, he’d been looking out for his friends and for a woman in unfortunate circumstances. The scandal was, of course, unfortunate, but the governor’s involvement was far from nefarious, and certainly shouldn’t preclude him from serving as president (especially not when Blaine had already made it clear he was not a man to be trusted).
Halpin's story was quite different:
In an October 31, 1884, interview with the Chicago Tribune, she proclaimed, “The circumstances under which my ruin was accomplished are too revolting on the part of Grover Cleveland to be made public.”
Halpin was a 38-year-old widow in 1874, according to the Tribune, which also reported:
Halpin said that Cleveland had pursued her relentlessly, and that she finally consented to join him for a meal at the Ocean Dining Hall & Oyster House. After dinner, Cleveland escorted her back to her boarding house. In an 1874 affidavit, Halpin strongly implied that Cleveland’s entry into her room and the incident that transpired there was not consensual—he was forceful and violent, she alleged, and later promised to ruin her if she went to the authorities.
Halpin said she told Cleveland she never wanted to see him again, but “five or six weeks later” was forced to seek him out because she was in the kind of trouble only Cleveland could help her with.
The trouble, of course, was pregnancy.
Nine months later, Halpin’s son was born and promptly removed from her custody. Halpin was admitted under murky circumstances to a local asylum for the insane. Doctors from that institution, when interviewed by the press during the 1884 campaign, corroborated Halpin’s insistence that she was not, in fact, in need of committing.