What are backlinks strategies for multi-location services?

Let's say I have 20,000 pages about babysitting: babysitting in new york, babysitting in LA, babysitting in Boston, ...

Do you guys have local SEO strategies to mass build backlinks to these pages?

Start building backlinks with those keywords and their LSI variations. Also exploiting Google local listing would be another great idea for such multilocation services. You will find such services in marketplace area, don't forget to search there as well.

You don't have to build backlinks to all 20000 pages you will build. Sending powerful backlinks to your homepage and having a proper SILO structure so that the link juice flows all over your pages would be the best solution in your situation.