Public APIs are widely used by application developers who integrate them into their applications. Developers often turn to online API support, such as the ones on Stack Overflow, in addition to official API documentation, when seeking help. There is a need to analyze and characterize API support problems in this environment. In addition, APIs operate in diverse environments, contributing to potential support problems. This study introduces a metric to classify public API support answers as faulty or not, based on downvotes on Stack Overflow. Furthermore, tree-based models are applied to analyze how key API environmental factors, such as integrated development environment, programming language, library, operating system, browser, and device, impact the occurrence of API support problems. Our analysis offers an objective assessment of API support problems and the environmental factors associated with them. This approach identifies and characterizes high-risk areas, or clusters of faulty answers, and provides insights that can potentially help API providers improve support strategies.