Ensuring the effectiveness of text-based crisis counseling requires observing ongoing conversations and providing feedback, both labor-intensive tasks. Automatic analysis of conversations---at the full chat and utterance levels---may help support counselors and provide better care. While some session-level training data (e.g., rating of patient risk) is often available from counselors, labeling utterances requires expensive post hoc annotation. But the latter can not only provide insights about conversation dynamics, but can also serve to support quality assurance efforts for counselors. In this paper, we examine if inexpensive---and potentially noisy---session-level annotation can help improve label utterances. To this end, we propose a logic-based indirect supervision approach that exploits declaratively stated structural dependencies between both levels of annotation to improve utterance modeling. We show that adding these rules gives an improvement of 3.5\% f-score over a strong multi-task baseline for utterance-level predictions. We demonstrate via ablation studies how indirect supervision via logic rules also improves the consistency and robustness of the system.