If bitcoin fails as a currency, how will blockchain tech still be secure for other applications? We often hear proponents, like Andreas Antonopoulos, say things along the lines of Bitcoin being an application of the Blockchain technology and that Bitcoin as a currency can fail, but blockchain technology will continue.

But the blockchain is only secure (and therefore useful) if there is a large enough and diverse enough (e.g. no one with >50% of the mining power) network of miners securing the network.

If the USD to BTC value goes down too low, or Bitcoin as a currency (for whatever reason) fails, the miners will stop mining. Is it therefore possible that this could then make the bitcoin blockchain technology insecure and put the other applications built on top of the blockchain technology at risk?

I guess one might say, what Andreas means is that new blockchain mining networks, independent of bitcoin could be setup. But what makes the bitcoin blockchain so good, is because the network of miners is so large (and therefore so secure), is it not?
I understand it exactly the same way as you .. the currency needs to be worth the time and effort of the miners or at least the have to believe it is in the long run.