The idea that Lightning will reduce the amount of fees going to miners is very common, and we probably won't know for sure until we actually try it. However, I'll try to provide some arguments as to why I don't believe it'll result in a drop in fees for miners, let alone a drop in hashrate.
I think the idea that Lightning will drain transaction fees from the Bitcoin network is wrong. First of all Lightning will enable completely new use-cases for Bitcoin, increasing its reach and resulting in a vast increase in the transaction rate and tapping into markets that are currently simply too expensive to serve. Every single one of these transfers will result in some (small) fees for the channel operator, which she can then use to pay for the on-chain transaction for the maintenance of the channel. The fees can be much higher than a classical Bitcoin transfer since the utility the operator derived from the coins in the channel are much higher, since they have been sent back and forth an arbitrary number of times.
Think of Lightning as an aggregation layer for Bitcoin: it processes transfers that the current Bitcoin cannot process itself, accumulates fees and then releases some of those fees back to the Bitcoin layer.
Secondly, Lightning shifts the security model from a model where we broadcast a transaction and eventually it'll confirm, to a model in which the confirmation time becomes time-critical. There are transactions in Lightning, which we need to get confirmed before a timeout, otherwise we risk losing some coins, due to an old state being committed to the blockchain. This time-critical confirmation encourages channel operators to attach larger than usual fees to these settlement transactions. Fees that end up with the miners.
Finally, on-chain Bitcoin and off-Chain Lightning transactions have different tradeoffs, I don't see the world becoming Lightning only, but I do hope that Lightning is a useful tool for some use-cases. In the end Lightning is a system that is built on Bitcoin, and thus can only be as secure as Bitcoin itself.
(disclaimer: I am one of the developers working on Lightning)