Multimodal learning, while contributing to numerous success stories across various fields, faces the challenge of prohibitively expensive manual annotation. To address the scarcity of annotated data, a popular solution is unsupervised domain adaptation, which has been extensively studied in unimodal settings yet remains less explored in multimodal settings. In this paper, we investigate heterogeneous multimodal domain adaptation, where the primary challenge is the varying domain shifts of different modalities from the source to the target domain. We first introduce the information bottleneck method to learn representations for each modality independently, and then match the source and target domains in the representation space with correlation alignment. To balance the domain alignment of all modalities, we formulate the problem as a multi-objective task, aiming for a Pareto optimal solution. By exploiting the properties specific to our model, the problem can be simplified to a quadratic programming problem. Further approximation yields a closed-form solution, leading to an efficient modality-balanced multimodal domain adaptation algorithm. The proposed method features \textbf{B}alanced multi-\textbf{o}bjective \textbf{o}ptimization for \textbf{m}ultimodal \textbf{d}omain \textbf{a}daptation, termed \textbf{Boomda}. Extensive empirical results showcase the effectiveness of the proposed approach and demonstrate that Boomda outperforms the competing schemes. The code is is available at: https://github.com/sunjunaimer/Boomda.git.