Yes, indeed, even an infant becomes impure.
The mishna in Taharos 3:8 says
תינוק הנמצא בצד העיסה ובצק בידו רבי מאיר מטהר וחכמים מטמאין שדרכו של תינוק לטפח
A child/infant found next to dough, and there is dough in the hand of the child/infant, R' Meir declares it pure and the Sages declare it impure, as it is the normal practice of a child to pat things.
Rabbeinu Tam (in Kiddushin 80a) cites the Tosefta that the reason the child is assumed to be impure, and therefore impurifies the dough if it touched it, is because Niddah women handle the children:
מפני מה אמרו שהתינוק טמא משום דנשים נדות מגפפות ומנשקות אותו
For what reason did [the Sages] say an infant is impure? Because Niddah women handle and kiss them.
The implication of the Tosefta is that they did not bother to remove them from their state of impurity. There was nothing forbidden about leaving them impure, one would just have to keep them away from things that were being maintained as ritually pure, such as terumah.
In terms of how we deal with it today, we don't! Since today we are all assumed to be impure with a much more serious impurity of corpse-impurity, from which we lack the current means to become pure, we are largely unconcerned with matters of ritual impurity, and do not address the Niddah impurity except for the sake of permitting a woman to her husband. There are few contemporary ramifications of impurity, such as for Kohanim to avoid corpse-impurity and to not cause terumah to become impure, but it is largely inapplicable for us in our current state.
