Tl;Dr you want your PCB assembler to have ISO9001 and IPC-A-610 certificates and IPC-600 from the PCB fab. These have to do with the supplier actually supplying a product that works, you would be ultimately responsible that you've done due diligence that the product is manufactured to your specifications.
IEC60950 requires a battery of tests to show it's safe, most of which you can actually sidestep by using low voltage (e.g. 24V) external power supply that's IEC60950 rated. You also need to contend with IEC61000 series of standards on RF emissions and immunity but that's another story.
IEC 60950 (now superseded by IEC 62368-1) is a top level safety standard that specifies a range of requirements and tests required to put the CE mark on a product signifying it's safe to use and to sell on market.
Having a proper QA system such as ISO9001 is often part of this but not absolute requirement depending on the exact product nature. You really do want an assembler/fab to have one of these however.
IPC-A-610 is a quality standard of PCB assembly. In basic level it's a standard you can quote to the assembler and they cannot weasel out of rubbish assembly job. There are three classes, "1" is for general equipment but to my mind it's borderline functional, larger misalignments are allowed than you might expect. "2" is for Dedicated installation electronics e.g. customer access portals and the like that require some fault tolerance. This is what I'd usually ask for. Class "3" is for high reliability products, it's nice to have but often suppliers balk at this.
If the supplier does quote this, they usually also have ISO9001 in place and maybe IPC-600 if they also make the PCBs.