Republicans most certainly. Between
Trump wanting a military presence on the southern border, and
the right to militarily intervene in Mexico should the need arise, it is pretty clear where this debate is headed. I think the hardliners just quit dancing around this like a horse and state flat out that they want a Hadrian's Wall (or the Walls of Constantinople to use another analogy) on the southern border, and that they are willing to invest the resources necessary to realize this vision, even if it means raising taxes.
It would almost certainly take care of border security, and would delineate the boundary between civilization to the north of the border, and those uncouth lawless barbarians to the south of it. This border 'wall' is nothing more than a glorified fence (and a campaign stunt) that anyone with access to metal cutting tools can overcome, and not worth the resources needed to maintain it. If you
really want to get serious about building a wall across the border, then build a proper wall above
and beneath the surface worthy of the name -and invest the resources needed for it, along with establishing permanent military bases set at regular intervals along its length to house those needed to garrison it -and to launch periodic military campaigns into northern Mexico.
I wouldn't advise trying to build the modern equivalent of the Antonine Wall deep in Mexican territory though.