Satellite internet serves as an indispensable component of the upcoming sixth-generation networks for providing global broadband internet access service. Due to the open nature of satellite-ground communication, security issue in satellite internet has always been an important concern for both industry and academia. Although many researchers focus on secure communications in satellite internet, the literature is surprisingly sparse, with no comprehensive overview of the state-of-the-art security techniques. This paper provides an in-depth survey of secure communications for various satellite internet scenarios. Based on different security mechanisms, we first categorize the existing works of secure communications in satellite internet into two categories: cryptography-based and physical layer security-based. The former includes classical encryption-based and quantum encryption-based secure communication, and the latter is further divided into precoding-based, cooperative jamming-based, relay selection-based, and physical-layer authentication-based secure communication depending on the applied techniques. Finally, we provide some future research directions.