Is it common for residential foundations from the 1960s to contain asbestos?
For concrete foundations no, asbestos is not an issue. However, the earth in your location may contain asbestos. In Northern California my brother is having a pool installed and they had to monitor for asbestos while digging.

The dust masks are used for cutting concrete or just the dust. The concrete and rock dust is not good to breathe so we usually wear masks.