Did Obama encourage illegal aliens to vote? Yes and no.
Per the not-so-credible fact-checking site Snopes , Hoft and Cavuto are wrong.
As you might suspect, Hoft, Cavuto, Drudge, and Snopes are all wrong. While Obama himself didn't encourage illegal aliens to vote, Rodriguez clearly wants some illegal aliens to vote and Obama didn't correct her explicitly. That might give some in the target audience the impression that Obama wants illegal aliens to vote.
However, Obama didn't explicitly say illegal aliens should vote as Hoft and Cavuto would have it. The Hoft post is at and, as has happened in so many other cases, it uses a video snippet of the video (not the full interview video below) to deceive his readers.And, that's important because as long as those who claim to oppose illegal immigration keep making false claims - even if they're partially correct - it just helps "fact checkers" discredit the opposition to illegal immigration. If those like Hoft, Cavuto, Drudge, etc. could learn to just tell the truth and the whole truth for once, they wouldn't make it so easy for the pro-illegal immigration side. Of course, telling the truth rather than making wild claims would probably impact their page views, so don't look for that to happen any time soon.
The transcript provided by Snopes, augmented by preceding parts of the interview, is below (bolding added). For those not familiar with these issues, the "Dreamers" Rodriguez refers to are illegal aliens. There's no such thing as her "undocumented citizen": those are also illegal aliens. Rodriguez clearly wants them to vote since she refers to them being "fearful of voting". It's less clear if she's then referring to them voting or to her (she's an American citizen) voting.
In Obama's answers, he clearly only refers to citizens voting. However, if he were an opponent of illegal immigration and illegal aliens voting, he would have clarified the ambiguity Rodriguez clearly wants beyond simply saying "[w]ho can't legally vote". And especially after saying that anyone who shows up at a polling place will be allowed to vote, and Rodriguez' reaction. Imagine how someone like Tom Tancredo or Jeff Sessions would have responded to the same questions. Whether Obama stoked ambiguity because he does in fact want illegal aliens to vote but can't say it, or because he's just too wimpy to engage a pro-illegal immigration advocate in how she's wrong isn't clear.
Note also Obama's obsession with us-vs-them and shadowy powers oppressing the people: "people try to discourage you from voting", "they can't stop you from voting", "the powers that be", and "they don't want people voting". That's almost like something someone in the Alt Right would say, except with them we know who they're referring to. Who might Obama be referring to, given that those with actual power in the U.S. are those on Obama's side of things?
OBAMA: ...Here in America there is a tradition of making it hard for people to vote. Historically, it was done purposely to prevent African-American from voting. And, that tradition has sometimes carried over. But, here's the good news. Even though people try to discourage you from voting, if you want to vote, and you show up at your polling place, they can't stop you from voting.
RODRIGUEZ (looking at camera): Hmmm!
OBAMA: We give our power away all the time. The powers that be, they're counting on us giving our power away.
RODRIGUEZ: Many of the millennials, Dreamers, undocumented citizens -- and I call them citizens because they contribute to this country -- are fearful of voting. So if I vote, will immigration know where I live? Will they come for my family and deport us?
OBAMA: Not true. And the reason is, first of all, when you vote, you are a citizen yourself. And there is not a situation where the voting rolls somehow are transferred over and people start investigating, et cetera. The sanctity of the vote is strictly confidential in terms of who you voted for. If you have a family member who maybe is undocumented, then you have an even greater reason to vote.
RODRIGUEZ: This has been a huge fear presented especially during this election.
OBAMA: And the reason that fear is promoted is because they don't want people voting. People are discouraged from voting and part of what is important for Latino citizens is to make your voice heard, because you're not just speaking for yourself. You're speaking for family members, friends, classmates of yours in school...
RODRIGUEZ: Your entire community.
OBAMA: ... who may not have a voice. Who can't legally vote. But they're counting on you to make sure that you have the courage to make your voice heard.
 In "Criminal President Obama Encourages Illegal Aliens to Vote - Promises No Repercussions (VIDEO)", Hoft writes "Barack Obama openly called on illegal aliens to vote in Tuesday's election." The video Hoft provides is Cavuto playing a snipped of the full video.
thegatewaypundit . com/2016/11/
 In "It Was Aliens: In an interview aimed at a millennial audience, President Obama encouraged Latino citizens to vote. He did not urge undocumented immigrants to vote", Bethania Palma says Hoft's claim is "FALSE": "Many of the outrage posts were written around a video in which the majority of the president's response was edited out to give the misleading impression that Obama was urging undocumented immigrants to vote illegally: in the full interview, it's clear Obama is urging Latino citizens to vote in order to give voice to members of their community who are precluded from doing so by lack of citizenship, not urging non-citizens to vote illegally."
snopes . com /