The Republican Party did generally have a close and sympathetic relationship with businesses, but it wasn't a total slave to big business and it did have other electoral concerns. For example, Harrison also signed into law the Sherman Silver Purchase Act and one of the first major conservationist laws, the
General Revision Act.
By the late 19th century, there was a widely-held notion that some businesses (particularly railroads) had gotten a little too big and powerful and that the federal government should step in to do
something. The Populist Party and its antecedents were calling for more far-reaching reforms, but two fairly weak laws were passed instead: the Interstate Commerce Act of 1887 (passed by a Republican Senate and a Democratic House, and signed into law by Cleveland) and the Sherman Act.
Ultimately, the Sherman Act was largely rendered toothless due to weak presidential enforcement (by Harrison, Cleveland, and McKinley) and Supreme Court decisions like United States v. E. C. Knight Co. It wasn't until Teddy Roosevelt took office that the U.S. got serious about breaking up trusts.