To determine the correct objective of using audio-lingual aids in a classroom, let's analyze each given option:

1. Spelling and punctuation skills: Audio-lingual methods primarily focus on listening and imitation rather than visual components like spelling and punctuation. Hence, this is not the primary objective.
2. Reading and writing skills: The audio-lingual method emphasizes oral-aural practices focused on listening and speaking, not on reading and writing, which often rely more on visual engagement with the text.
3. Listening and speaking skills: This is the primary focus of audio-lingual methods. The technique is based on repetitive listening and speaking exercises to internalize language patterns and develop fluency.
4. Vocabulary and grammar: While vocabulary and grammar are important, the audio-lingual approach uses them as tools in the service of developing practical listening and speaking skills.

Given the nature of the audio-lingual method, which is grounded in the principles of behaviorist language learning (repeated practice, positive reinforcement, etc.), the prime objective is indeed to develop listening and speaking skills.

Conclusion: The correct answer is that the prime objective of using audio-lingual aids in a classroom is to keep students developing listening and speaking skills. This method effectively enhances students' ability to comprehend spoken language and articulate responses in the target language.