At the moment, the Holocene extinction fails to meet the criteria for a mass extinction. In a mass extinction 75% of species go extinct in a short space of time. However, knowledge of memetics suggests the possibility of a memetic takeover - in which the substrate of the heritable medium of biology changes - and the era of the DNA molecule comes to an end. Such an event could be accompanied by a large mass extinction. In such circumstances, many DNA-based creatures would probably be preserved in nature reserves or historical simulations - and thus not technically go extinct. However, it is far from clear what fraction of the currently-known species would survive.
One problem with extinctions is that things of value get lost. Assuming that preventing this is desirable, we can ask what steps could be taken to help prevent species loss.
I think a slower memetic takeover would result in a reduced chance of things getting lost by accident. Technological determinism probably means that we can't do much to slow progress by machines down. However, what we probably can do is speed up our own progress - and that of our companion creatures. The aim would not be to keep up with the machines - which does not look feasible - but rather to prolong the period of our usefulness - and prevent us from becoming an early casualty of the current extinction.