Miners have basically no choice other than to accept it if they want to keep mining a solid and only increasing coin. If they don't agree with this off chain scaling solution, they are free to leave or to fork off. As we speak, miners don't need the transaction fees as long as the price remains at least close to current levels. Best of all is that lightning network isn't something you are forced to use. If you don't want to, you can just keep transacting like you always have been doing -- you can see it as an optional feature. On top of that, opening and closing channels require fees too, so it's not that miners won't get any fee income anymore. Combine that with people transacting in the non lightning way, and the miners likely won't even notice much of a difference.
Besides, mining should not be the primary objective of the Bitcoin system. Miners are simply a means to an end. Ideally, they would participate if and only if the current fees are enough to offset the costs required of them to mine given the cost of the hardware, the cost of the electricity, and the value of Bitcoin itself. If mining ever becomes not profitable for any one miner causing them to stop, the difficulty will decrease and other miners will benefit more. If at any point there is a majority group of miners, it most likely means that the difficulty is too low and that mining is very profitable, causing more miners to come and compete. It is quite a well designed system with ties to supply and demand theory. Those who can profit from it should do it and those who cannot should not.
And it seems that for the last two years miners have been an obstacle to Bitcoin adoption and development. Instead of helping Bitcoin in every possible way, the miners have been actually hampering its progress. Their economic interests don't quite match the interests of the Bitcoin community and Bitcoin itself, so we shouldn't actually blame them for that. I guess anyone in their place would do the same, that is working the transactions fees up no matter what.