In terms of mankind's presence passively helping another species achieve sapience, there probably wouldn't be any species that fit your criteria. Humans haven't been creating strong selection pressures for higher intelligence, and the only way our presence would help another species achieve sapience is if we outright uplifted them. There have been suggestions that the complex artificial environments created by humanity have selected for larger brain sizes in squirrels and such, but it's not going to act on them to the point that they develop sentience. Indeed, in the absence of humans it is possible that these squirrels might lose any increase in brain mass, in a similar case to how the peppered moth returned to a non-black morph when industrial pollution levels fell.
The species that will benefit evolutionarily will primarily be invasive species mankind introduced to isolated landmasses, such as islands. By introducing these organisms to new landmasses, mankind has increased the total range available to these animals, created new opportunities for speciation, and reduced the chance that these species will go extinct by creating potential refuge populations where they could continue to adapt and thrive (especially on islands). Many groups of animals, like ungulates and carnivorans, are normally terrible at crossing water barriers, and humans represent an unprecedented opportunity for them to get to landmasses they would otherwise be unable to. Hawaii, New Caledonia, and New Zealand are good examples of this, as non-flying mammals would otherwise have no chance of reaching these landmasses for over 100 million years. Although some of these islands, like Hawaii, would eventually sink beneath the waves, many species could still keep island hopping the way Drosophila did in Hawaii and some islands such as New Caledonia, Madagascar, and New Zealand are actually continental fragments and would last a very long time.
There are some qualifiers to this. Although many species in Eurasia have been introduced to North America and vice versa, many of these species would have gotten to these locations anyway the next time the Bering strait closed and species could freely migrate between the Old and New Worlds. For example, there are no native true rats (Rattus) in North America, but there are plenty in Asia and they would have just crossed the strait some time in the next hundred thousand years or so.
Human-introduced species are a big deal compared to other species that currently benefit from human aid (livestock, pests) because while it took human intervention to get them there, human intervention isn't required for them to survive there. Which means that if humans go extinct these new, invasive populations won't go extinct with them (this is the problem with things like cockroaches).
Given these factors, I would say the species that has benefitted the most from the presence of humanity (considering their current distribution and how extensive their distribution was pre-humanity) would have to be...
Rainbow Trout (Oncorhynchus mykiss) and Goldfish (Carassius auratus)
Freshwater fishes have a much harder time dispersing to new habitats, since they are dependent on either shifts in river flow, floods to overcome barriers like waterfalls, or travelling through the ocean to get to new rivers. Even catadromus/anadromus species can be blocked by continental landmasses or mountain ranges. Rainbow trout are native to the Pacific Coast of the U.S. and Canada and the Kamchatka Peninsula in Russia, but they have been introduced absolutely everywhere, including eastern North America (which it wouldn’t normally be able to get to due to the Panamanian land bridge and the Rockies), Eurasia, South America, Africa, Australia, New Zealand, and even Hawaii. They are now on every continent and most major land masses except Antarctica. That is a huge expansion of their range, and they would not have been able to get to many of these places naturally. Brown trout (Salmo trutta) have a similar history. In some places these trout are artificially bred, but in most they have breeding populations.
Goldfish are kind of similar. They’re native to eastern Asia but have been introduced just about everywhere as a consequence of their status as common pests, and got to places like North America where there were no native carp beforehand. Common carp are less so because they already had a large range prior to human introduction, humans have not benefitted them as much.
And a terrestrial runner-up would be...
House cats (Felis catus)
Species of Felis are native to most of Africa, Europe, and Asia, but humans have introduced them to every major landmass and most islands except Antarctica. Cats aside from tigers are normally terrible at crossing water barriers due to their hypercarnivorous habits. Additionally, compared to other domestic animals (except maybe goats and dogs), cats are much better at surviving in the absence of humans. Humanity's presence has single-handedly given cats a whole bunch of evolutionary opportunities they normally wouldn't have.
Rats (Rattus spp.) are a potential third place, but rats are already good at dispersing to places and there are already native rat species in far-off places like Christmas Island, the Phillipines, and Australia long before humans came along. Getting to places like the Galapagos or Hawaii by human introduction is notable, but overall the species hasn't benefitted long-term as much as cats have.