Every season the American Red Cross runs Baby Sitting certification classes in most communities. The fee is less than $20 (or free in our town as the town provides the space and tuition as a community service)
We had our two youngest take this course and felt much more comfortable allowing them to babysit after formal training as to what steps to take in an emergency, what is proper behavior while babysitting, basic first aid response, etc.
In addition to references, I’d insist on certification/formal training before engaging the services of a baby sitter for my grandchildren.
This past week when we hosted my daughter’s chassunah, I hired 3 graduates of the spring class to supervise the youngest guests so their parents could enjoy the simcha weekend.