Phase-modulated continuous-wave (PMCW) radar is one of the emerging technologies in various wide-band applications, such as automotive radars. Utilizing a one-bit analog-to-digital converter (ADC) at the receiver side can reduce the cost and power consumption of this kind of radar while leveraging its benefits. In this correspondence, we consider the joint design of the binary transmit code and the receive filter for PMCW radars in the presence of a one-bit ADC at the receiver. When using a one-bit ADC, noise samples play an essential role in the sidelobe increment of the filter output. Therefore, the mean-integrated sidelobe level (MEISL) metric is used as the design criterion to cover the effect of noise. Since the target backscattering coefficient ($\alpha$) is unknown at the design stage, we resort to the worst-case approach concerning the target backscattering coefficient and cast a minimax problem. We demonstrate that the inner maximization problem has a closed-form solution. Indeed, we prove that the maximum value of MEISL for the fixed code and filter occurs at a real-valued $\alpha$ that relates to the minimum acceptable signal-to-noise ratio (SNR) of the radar system. Finally, by recasting the objective function, a cyclic optimization procedure is utilized, for which the filter optimization subproblem has a closed-form solution, and the code is optimized using the coordinate descent framework. Several numerical examples are provided to evaluate the effectiveness of the proposed design procedure. The proposed method exhibits superior performance compared to some state-of-the-art methods, particularly in low-SNR scenarios.