Intelligent reflecting surfaces (IRSs) enable multiple-input multiple-output (MIMO) transmitters to modify the communication channels between the transmitters and receivers. In the presence of eavesdropping terminals, this degree of freedom can be used to effectively suppress the information leakage towards such malicious terminals. This leads to significant potential secrecy gains in IRS-aided MIMO systems. This work exploits these gains via a tractable joint design of downlink beamformers and IRS phase-shifts. In this respect, we consider a generic IRS-aided MIMO wiretap setting and invoke fractional programming and alternating optimization techniques to iteratively find the beamformers and phase-shifts that maximize the achievable weighted secrecy sum-rate. Our design concludes two low-complexity algorithms for joint beamforming and phase-shift tuning. Performance of the proposed algorithms are numerically evaluated and compared to the benchmark. The results reveal that integrating IRSs into MIMO systems not only boosts the secrecy performance of the system, but also improves the robustness against passive eavesdropping.