While there is no clear, direct, 'Influence Meter', or anything so crude in the game, there is one reasonably reliable metric for measuring influence - Alignment.
Companions with whom you have high influence will tend to mirror the alignment of the Exile - this is particularly easy to see with a dark side Exile; as your influence rises, the backdrop of the portraits of your various companions on their character sheets will get darker and redder, reflecting their fall to the dark side with you.
(N.B., some companions, such as Kreia, will never shift alignment, and since most companions start out at least somewhat on the light side of the spectrum, it's a bit tougher for a light side Exile to notice the shift.)
Also, with negative influence the Exile's companions will align with the opposite polarity. Meaning, if you lose influence with anyone as a dark side character they will align with the light side to oppose you. Gaining negative influence also unlocks conversations and the ability to create jedi like positive influence does.