how can it be such people are considered "illegal" even though their presence here is no fault of their own? What "crime" did they commit?
They're "illegal" in the sense that they don't (prior to any amnesty being granted) have any lawful right to reside in the USA, but they are residing in the USA. "Legal" and "not legal" aren't matters of "fault", they're matters of legal definition. To slightly misapply criminal terminology, there is no mens rea for "not having legal status to reside in the US", it's a strict-liability thing.
Anyway, it's not the case that everything illegal is "criminal". Especially in US terminology, where "criminal" is often reserved to mean "felonious", therefore excluding "misdemeanour" crimes.
I'm not a lawyer, I'm not even a US resident myself, but for example one relevant law might be 8 U.S. Code § 1227 (a) (1) (A) "Any alien who at the time of entry or adjustment of status was within one or more of the classes of aliens inadmissible by the law existing at such time is deportable". This alone doesn't assert that they've committed a crime. That whole section of USC is about people who have no legal grounds to remain in the USA, since they legally can be deported. These people who never had residency or who had a time-limited residency which has expired, are often informally referred to as "illegals" regardless of whether they've committed any criminal offence that can be prosecuted.
In addition, they may actively have committed crimes once they reached the age of criminal responsibility, and/or once they reached the age of adulthood, and continued to reside in the USA without having legal status to do so. AFAIK it's not an offence merely to reside in the USA without status, so many or most "illegals" are not criminals in that sense, but a diligent prosecutor might be able to establish in particular cases that someone has "harbored" other illegal aliens or whatever. There could, I suppose, be offences related to employment or tax: it's certainly possible for an undocumented immigrant to pay tax, and many do, but many don't. So it would not be correct to assume than an "illegal" necessarily is a criminal.
So far as the law is concerned, deporting a person who has no legal status to reside is not punishment, at least no more than evicting someone from private land where they have no right to live would be "punishment". Deportation is not considered a criminal matter. So, those who oppose DACA do not agree that, when someone does not have right to reside in the USA, "the least that can be done" is to let them stay in the country. They think the least that can be done is to deport them or ignore them, and any more is a kindness that they choose not to extend.
Of course you are quite correct that many Dreamers have no practical ability to live productively anywhere else. But those who want them deported, or at least who believe that the state should reserve the right to deport them, do not necessarily feel that compassion towards non-US citizens is their highest moral imperative or political priority. Some proportion of them might feel that a person with no right to reside in the USA, is morally obliged to leave the USA at the first opportunity, regardless of their prospects in their country of nationality. However, that's not the legal situation.