The Farrow-structure-based steerable broadband beamformer (FSBB) is particularly useful in the applications where sound source of interest may move around a wide angular range. However, in contrast with conventional filter-and-sum beamformer, the passband steerability of FSBB is achieved at the cost of high complexity in structure, i.e., highly increased number of tap weights. Moreover, it has been shown that the FSBB is sensitive to microphone mismatches, and robust FSBB design is of interest to practical applications. To deal with the aforementioned problems, this paper studies the robust design of the FSBB with sparse tap weights via convex optimization by considering some a priori knowledge of microphone mismatches. It is shown that although the worst-case performance (WCP) optimization has been successfully applied to the design of robust filter-and-sum beamformers with bounded microphone mismatches, it may become unapplicable to robust FSBB design due to its over-conservativeness nature. When limited knowledge of mean and variance of microphone mismatches is available, a robust FSBB design approach based on the worst-case mean performance optimization with the passband response variance (PRV) constraint is devised. Unlike the WCP optimization design, this approach performs well with the capability of passband stability control of array response. Finally, the robust FSBB design with sparse tap weights has been studied. It is shown that there is redundancy in the tap weights of FSBB, i.e., robust FSBB design with sparse tap weights is viable, and thus leads to low-complexity FSBB.