We don't know, but looking at it objectively leads to some deeply unpleasant conclusions
An extensive review of the film and its additional materials reveals no obvious way that humanity has been able to survive. The demographics are just weird (loads of adults of breeding age but almost no children?) which is great from a story-telling perspective, but almost impossible to justify from a worldbuilding POV.
We could solve these problems by simply having thousands of passengers and hundreds of children elsewhere on the ship (e.g. that we simply didn't happen to see) but we know from the script that this isn't the case when the entire passenger complement is summoned to the lido deck in the final sequence.
What we know
Based on the captain's portraits, a healthy human can live as to be as old as 130. It's likely that women in the future are capable of breeding from the ages of 15+ to around 70+, giving a total 'lifetime window of fertility' of some 55 years.
The total number of adults seen (in the final sequence where all of the passengers gather on the Lido Deck) is less than 15,000, down from an original complement of 600,000. We know from an earlier answer that Auto is probably turning any dead bodies into food. But wait, it gets worse.
The total number of children seen on screen is approximately 30. It should be closer to 700, even assuming you only want to keep the population stable at 15,000
Everyone on board appears to be aged between 20-40 years of age. There are no old people and no teens.
Even if pod-breeding was possible, it's highly unlikely that a cruise ship could be retro-fitted with the hundreds of pods that would be needed to maintain a stable population without massive internal changes, none of which are obvious in the ship's deck map.
Conclusions (look away now, not for the squeamish)
We know that the passengers enjoy virtual dating and it's not hard to imagine that a good virtual date would end with virtual sex, affording the ship the opportunity to impregnate the woman with a viable sample of the man's sperm.
MARY: Date?! (derisive snort) Don't get me started! Every holo-date I've been on has been a virtual disaster! If I could just
meet one who wasn't so superficial. There are no good men out
Alternatively, the women on the ship are simply impregnated at an appropriate moment (again, by robots) in waves, thus allowing the ship to repopulate itself periodically.
The reason that the passenger complement looks so odd is that we're looking at a quiet period where Auto is only breeding small numbers of humans, presumably to rebalance the number of male and female births from the last big fertility "wave" and replace any children that they've had to euthanise due to birth defects. Obviously anyone over breeding age is also worthless to the ship, and again it's likely that they've either been dumped overboard or mulched up into soylent-green-in-a-cup.