In this era of massive data, DNA serves as a promising new medium for information storage. Compared to traditional physical storage media, it possesses inherent advantages such as low energy consumption, high storage density, and long storage lifespan. With the rapid development of DNA storage, ensuring information security under new technologies becomes crucial. In this regard, this paper combines research in the fields of encryption and DNA coding, proposing a DNA encryption coding method based on chaotic systems and fountain codes. The encryption principle of chaotic systems is utilized during the DNA fountain code encoding process, preserving the characteristics of DNA fountain codes while ensuring the security of encoded information. This method is applicable to any types of data, achieving high information density and DNA encoding under arbitrary constraints. Furthermore, through simulation experiments, it is demonstrated that this method can effectively resist various cryptographic attacks and possesses error-correction capabilities for data errors generated during the DNA storage process.