Basic beliefs are beliefs that are assumed, not inferred. They do not require arguments to be rational, because they are immediate and direct. They are formed by cognitive faculties that are functioning properly and are operating in the appropriate environment.
These cognitive (or belief-forming) faculties would include perception (senses), reason, experience, introspection, memory, intuition, credulity (testimony), and sensus divinitatus, and we use them constantly to form new beliefs.
Examples of properly basic beliefs:
I see a tree (perception).
I think therefore I am (reason).
I am in pain (introspection).
I had breakfast today (memory).
The grass is wet because it rained (intuition).
I was born in 1988 (credulity/testimony).
God exists (sensus divinitatis).
This doesn't mean that basic beliefs cannot be challenged or proven wrong. They can and should be revised or rejected based on new evidence (sometimes called defeaters).