It's multifaceted.

If no-kill shelters ever make sense (and some people must think they do, as these are things that do exist), then no-kill humanitarian efforts also must make sense. I find this to be self-evident because I think that humans are more important than domestic animals are.

UBI directly helps with the latter; UBI helps people avoid death.

UBI also indirectly helps with the former: By reducing the burden of taking care of pets when the bottom starts to fall out on folks' lives for whatever reason that happens, the need for dog shelters of any sort is also reduced.