Asylum-seekers are handled differently, these are people trying to improve their own finances by working for less than is legal, off the books. That almost entirely helps the rich and corporations who hire them, slightly helps themselves, and is done at the expense of actual citizens (including legal immigrants!!)
It's persistently a net economic boon, and their stress on public services are notably less than citizens or documented immigrants, rol. As well as being a boon they're willing to uproot their lives and then risk life and limb for.* By and large it ain't costing us shit except for the abuses coming out the companies exploiting it, and it damn sure ain't our enforcement efforts that are improving the situation on that front. Rather the opposite, 'cause I guaran-damn-tee you those folks like little more than a prospective employee unwilling to go to the cops or lawyer up for a wage dispute.
If our asylum seeker system was itself worth a shit, we'd have solid reason to mention it, but, y'know, like other parts of our immigration system it ain't worth near as much as it damn well should be. Maybe could be without the anti-immigration folks doing everything they can manage to fuck immigration legal and otherwise, but that ain't what we're dealing with right now.
The exploited population ain't who we should be shitting on, here, basically, even if they are fitting into below table economies (not that that shit isn't plenty common elsewhere, yet oddly no one seems to raise a fuss about it). They're neither the primary cause, nor is what they're doing morally questionable (well, unless you're on board with pissing on the however many times great grandparents of what's close enough to the entire country it barely matters, or chunks of our country's citizenship, too), particularly to any extent worth a damn.
Shit on what's making things bad enough they're seeing cause to leave, shit on the businesses giving incentive to come (or hell, figure out a way to make it less attractive for the employers in question, preferably without causing everything else involved to come out worse), shit on the people making it a nightmare to come legally, and so on. Once all that mess is dealt with, folks still trying to dig through holes can be something to fling the next round of shit at.
* This, incidentally, is the heroic bit, to the extent anything is. It's considered that because it's seriously the exact goddamn thing most of our ancestors did, or close enough the only people it matters to are those actively trying to shit on people doing what they think is their damnedest to improve the lives of them and/or their family.