Most existing studies realize bearing fault diagnosis tasks in labs with weak noise. However, field noise is so heavy under actual conditions that some methods may suffer from degradation or failure. To solve this problem, a fault diagnosis framework is proposed based on synchrosqueezing wavelet transform and kernel extreme learning machine (SWT-KELM). First, vibration signals are collected, and white Gaussian noise is added. Second, SWT is employed for signal decomposition in the time-frequency domain, and inverse SWT (ISWT) is applied for sub-signal reconstruction. Sub-signals with high correlation coefficients are selected for further feature extraction, specifically by singular value decomposition (SVD) to obtain singular values as the fault feature. Third, the KELM model, in which the beetle antennae search algorithm (BAS) is employed for parameters optimization, is constructed to classify the faults. For verification, the proposed method is implemented on the Case Western Reserve University (CWRU) dataset and Lab-625 dataset, and the results show that it maintains satisfactory outcomes on original and noise-contaminated data. Specifically, under noise conditions, the accuracies of the two datasets average at 96% and 83%, respectively, indicating the robustness and generalization of the method compared to other methods.