I'm at a keyboard now.
Cosmology doesn't have a direct answer to this. Direct observation into the past (by looking out to large redshifts) is limited by the time of last scattering, beyond which the universe was opaque. Everything beyond the surface of last scattering is extrapolated with well-understood mechanisms (some of which have made some of the best predictions to match experimental observation in the history of science).
So we already have the problem that there isn't direct observation, even though looking "out" is looking "back."
The other problem is that we do not have a working theory of quantum gravity, which would be necessary to really understand what was going on when the universe's size was on the order of the Planck scale. This includes understanding what would have been occurring around the Planck time after the Big Bang event.
Since time's arrow is given by the entropic gradient, and it's questionable what sort of entropic events would be happening on the order of one Planck time, it's not even really clear whether the concept of having a "time before the first Planck time" even has cognitive meaning.
This is a really roundabout way of saying "there isn't sufficient data, and the question might be wrong in the first place." Asking what was "before" the Planck time may well be like asking what's north of the North Pole.
Having said that, though, there is the concept of metatime. Temporal dimensions are always characterized by gradients, and there may be other gradients by which to give time an arrow than this universe's entropic gradient. For instance, the best way to understand this is by imagining a hypothetical multiverse where new universes are created: within each universe, there is a beginning to time (the entropic maximum for that universe), but overall there is a more encompassing metatime.
And this brings me to the most direct possible answer to your question: do we have good reasons to think there might be an infinite past?
The answer is "yes, but..."
The "yes" portion is that if inflation is true (and it has every appearance of being true), then it's really hard to avoid a multiverse because the inflaton fields decay asymmetrically, leading to "bubble universes" where it has decayed (and ours would be one such example). While inflation decayed here, it is still ongoing elsewhere, and has no reason to ever end, and no reason to have ever began.
The "but" portion is that while inflation is a scientific inquiry in terms of the local universe, I wouldn't strictly call eternal inflation elsewhere scientific because we can't empirically observe it: we have to leave the realm of pure science and enter the realm of philosophy to make those kinds of assertions. I am not saying that means the assertions are without evidence or without good reasons to think them, however.
Let me wrap this already-too-long response up by saying this: what we can say for sure is that there is zero evidence that the universe ontologically began to exist, even with the Big Bang (the most we can say is that its present local state began). This means that it remains a possibility that the universe is infinite and eternal. I am not sure how that could be shown, however.
