Bile, Not Blood
Technically it was not the Hydra's blood that was venomous at all. The confusion regarding this seems to arise from the fact that it was Hydra-venom in combination with someone else's blood that eventually brought an end to Herakles the Hydra-slayer himself. There does not seem to be any ancient reference to Herakles using the Hydra's blood itself as poison, rather it was the monster's gall or bile which was venomous, and this is what he dipped his arrows in, and is what sealed his arrows as fatality weapons.
The only reference which comes close to implying that it was the dragon's blood rather than its bile is a cursory expression used in the Roman Seneca's play Hercules Furens ("Hercules' Madness"). In it, Hercules describes the arrows as dipped in Lernæa nece, which Frank Justus Miller translates as "Hydra's gore" but literally comes closer to "Lernaea's violent murder/slaying".
Herakles [Hercules] destroyed a good number of his enemies (and a few friends too) by the use of his poisoned arrows. One of the last of these enemies was the Centaur Nessos [Nessus], who tried to violate Herakles' bride Deianeira. As he was dying, Nessos thought quite fast about how he could be avenged on his slayer even after he himself was long gone. He convinced Deianeira that his own Centaur blood (in one version this is mixed together with another bodily fluid) was a philtron [love-potion] which might come in handy someday if Herakles ever became unfaithful to her, and Deianeira collected what she could of it as an insurance policy.
True to form, years later, according to Sophocles' Trakhiniai, Herakles decided to pursue an old flame of his. Deianeira then took the garment which Herakles planned to wear to his new wedding and dipped it in her stash of Nessos' blood. Nessos seems to have understood full well that the Hydra-gall would remain in his blood permanently (whether the blood was actually a philtron at all, which I suppose we'll never know). So now, years after the Centaur's death, as soon as Herakles put on the garment, the dragon's gall in Nessos' blood burned him so badly that he couldn't bear to live anymore.
As for Herakles' encounter with the many-headed serpent, Hyginus does indeed paint our hero into a difficult scenario. It must have taken quite a bit of heavily exerted breathing to repeatedly chop off several heads from a giant monster while also having to fend off a giant crab. If even the creature's own footprints were poisonous [whether the dragon was present or not], without some decent self-insulation, it really should be virtually impossible to survive an actual showdown with such a beast. There are two possibilities I can think of for how Herakles could have slain a massive killer snake in such a toxic environment, let alone survived said environment.
The Lion's Pelt
To Kill the Hydra was
Mission 2.0 on Herakles' eventual list of 12 Tasks that he had to complete as a slave of his kinsman King Eurystheus of Argolis.
Mission 1.0 had been To Kill the Lion of Nemea, which lion, according to Apollodorus, was a brother of the Hydra. At any rate Herakles did complete his first mission and from then on he wore the skin of the Nemean Lion as armour practically everywhere he went. The lion pelt was impenetrable and impervious, and in ancient artwork the hero is usually depicted wearing the skin while battling the Hydra. There is no explicit written reference to the idea that the skin might have shielded Herakles in any way against the dragon of Lerna but there are some clues in his later adventures as to how this might have been so.
Mission 12.0, the last of the Eurysthean slave tasks, was To Capture Kerberos [Cerberus], the monstrous watchdog of the Underworld (and yet another brother of the Hydra). According to Apollodorus' Bibliotheka, one of the reasons that Herakles survived this one is because he covered his own skin completely in the Nemean Lion's pelt, although we have neither more detail about this nor any artwork graphic enough to display for us precisely what this might have looked like. Apollodorus simply tells us that by doing this, Herakles was able to withstand the bites of Kerberos' tail, which was composed of tens of snake- or dragon-heads, in a structure which reminds of the Hydra.
It should be reasonable to assume that a many-headed giant snake would try to bite its attacker, so for sure Herakles' recently acquired armour would've come in handy for protection against wounding during his 2nd Task. However, from what we can see of both Ancient and Modern Greek depictions of him clad in the pelt, there is no obvious means by which his respiratory system was also protected.
The scholion on Iliad 23.841 says that when Aias [Ajax], the son of Herakles' nephew Telamon, was a baby, Herakles wrapped him up in the skin of the Nemean Lion so that Aias became invulnerable like his cousin Akhilleus [Achilles]. The idea here seems to be that an infant's coming into contact with the lion-skin caused the infant to become impervious, as opposed to an adult upon whom such "magic" clearly did not work the same way, since Herakles himself, who wore the pelt all the time, did not grow impervious skin from such extended periods of interaction therewith. Perhaps, though, there was some minor, less permanent effect on an adult whose head was helmeted inside the head of the flayed lion, in that it created a pocket of cushioned air.
In case that is not especially convincing (and I don't think that it is) a final insight on it is that both Seneca (again in Hercules Furens) and Hyginus, as well as Aelian, say that the Nemean Lion was the son of the moon-goddess. According to Hyginus, the Moon even suckled the lion. (The lion and Herakles had this in common: having drunk the milk of goddesses, since Herakles, when he was a baby, was once suckled by Hera, at least according to Pausanias.) Such contact with the gods often leaves mortals with the scent of ambrosia upon them, and ambrosia has curative and preservative properties. Hence its use by Aphrodite to embalm the corpse of the Trojan prince Hektor [Hector] while Akhilleus was angrily dragging it behind his chariot after slaying the prince. And Dionysos' granddaughter Hypsipyle inherited the wine-god's purple robe which, after two generations, still smelled of the ambrosial perfume which had been placed thereon long before she was born.
In short, it may be that the scent of the Nemean Lion protected Herakles, or something he drank when he was a baby augmented his respiratory immunity. Even if this is acceptable, there is, however, the problem of the fact that Herakles' charioteer nephew Iolaos was present at the battle against the Hydra. Iolaos was not wearing any magical prophylactic that we know about, nor had he benefited from any divine breastfeeding, but he evidently survived the encounter with the Hydra just fine.
Protected by Pallas
With that being the case, we are essentially left with artwork to rely on as the main explanation source for the dilemma. Hyginus, Valerius Flaccus, Pausanias and Hesiod make brief mention of the idea that Herakles' half-sister the goddess Athena masterminded the strategic aspects of Herakles' 2nd Eurysthean task, and, according to Hyginus, it was this goddess who instructed the hero to gut the dragon's carcass in order to harvest her gall for arrow-poison.
Ancient artists corroborate Athena's involvement here with their renditions of the story, in which it seems that Athena, the protector of heroes (especially if they're her own brothers like Perseus and Herakles), was standing behind Herakles throughout the Hydra battle.
She is the woman to the far left in the image above and in the one below.
Here she is even brandishing what looks like a spear with which she is either goading Herakles on or is ready to skewer the monster alive, before the hero even has a crack at it.
It may simply be that Pallas Athena, somehow, protected both Herakles and Iolaos from the Hydra's breath.
The generally accepted interpretation of the legend is that “the hydra
denotes the damp, swampy ground of Lerna with its numerous springs
(κεφαλαί, heads); its poison the miasmic vapours rising from the
stagnant water; its death at the hands of Heracles the introduction of
the culture and consequent purification of the soil” (Preller). A
euhemeristic explanation is given by Palaephatus (39). An ancient king
named Lernus occupied a small citadel named Hydra, which was defended
by 50 bowmen. Heracles besieged the citadel and hurled firebrands at
the garrison. As often as one of the defenders fell, two others at
once stepped into his place. The citadel was finally taken with the
assistance of the army of Iolaus and the garrison slain.
HYDRA in Vol. 14 of The Encyclopaedia Britannica: A Dictionary of Arts, Sciences, Literature and General Information (11th Edition, 1910-1911)