The word for ‘base’ is ‘āyatanaṃ’, which means ‘sense object’.
The base or sense objects of infinite space & infinite consciousness are two of four very advanced & subtle meditation states, formally called ‘arupa jhana’.
For example, the most basic stage of meditation is experiencing breathing (& the body). When the breathing calms to a degree so it is no longer discernible by consciousness, rapture & happiness (pleasant feelings) become the object of meditation. When pleasant feelings calm/dissolve, the feeling of equanimity becomes the object of meditation. When the feeling of equanimity dissolves, infinite space becomes the object of meditation. When the space dissolves as the object of meditation, consciousness itself becomes the object of meditation.
Something like this. This process of mental purification can be read about in MN 111.
As for the term ‘ordinary person’, in MN 1, it refers to the unenlightened mind that attaches to anything (including infinite space & infinite consciousness) as ‘I’, ‘me’, ‘mine’ or ‘self’.
How exactly is consciousness infinite ? What does that mean?
“Bhikkhus, to whatever extent I wish, by completely transcending the base of the infinity of space, aware that ‘consciousness is infinite,’ I enter and dwell in the base of the infinity of consciousness. Kassapa too, to whatever extent he wishes, enters and dwells in the base of the infinity of consciousness. https://suttacentral.net/en/sn16.9/6
Usually, the mind & consciousness have a sense object, such as breathing, feelings, etc. It sounds like consciousness itself becomes the dominant sense object. If we get there, we might know for sure rather than speculate!