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Clinical trials in the medical domain are constrained by budgets. The number of patients that can be recruited is therefore limited. When a patient population is heterogeneous, this creates difficulties in learning subgroup specific responses to a particular drug and especially for a variety of dosages. In addition, patient recruitment can be difficult by the fact that clinical trials do not aim to provide a benefit to any given patient in the trial. In this paper, we propose C3T-Budget, a contextual constrained clinical trial algorithm for dose-finding under both budget and safety constraints. The algorithm aims to maximize drug efficacy within the clinical trial while also learning about the drug being tested. C3T-Budget recruits patients with consideration of the remaining budget, the remaining time, and the characteristics of each group, such as the population distribution, estimated expected efficacy, and estimation credibility. In addition, the algorithm aims to avoid unsafe dosages. These characteristics are further illustrated in a simulated clinical trial study, which corroborates the theoretical analysis and demonstrates an efficient budget usage as well as a balanced learning-treatment trade-off.