In some places, perhaps many, loss of biodiversity may have triggered ethoi (the plural of ethos) of conservation, where taboos, totemic species and other institutions helped preserve a minimum suite of species and reduce Tragedies of the Commons.
As for health, there are some examples (e.g. Lyme disease, malaria, Ebola and most recently Zika) where changes in biodiversity and ecosystem function have modified infectious disease risk, but relationships are bi-directional and disease and context specific. More straightforward, utilitarian arguments exist seeking to preserve species as reservoirs of as yet undiscovered compounds with pharmaceutical and other human benefits. Others advance ethical reasons, especially to conserve charismatic animal and plant species.
Unfortunately, none of these arguments, even combined, are sufficient to slow the juggernaut of biodiversity-eroding economic “growth”. Those with high biophilia have to think even more broadly, co-operating even more intensively with others who strive for global social and ecological justice, a strengthened eco-noösphere and a reduced human footprint on Earth.
PS Added July 22, 2016.
This was a talk to a fairly influential audience, about 800 people. In Thailand, where the meeting was held, criticising the royal family means prison. I very cautiously raised three major issues: 1. Human population increase; 2. Corruption (eg resulting in clearing of forests for palm oil, with trampling of the rights not only of local species such as orangutans but also of some Indigenous groups); 3. there needs to be a campaign to attempt to embarrass (stigmatise - though I did not say this) the consumers of products whose production is based on cruelty and biodiversity harm; similar to the antif-fur campaign; eg my slide on rhino horns: what I really wanted to suggest was that men using rhino horns as an aphrodisiac might be better off chewing their fingernails .. instead I meekly called for "more evidence" of benefit.