If there are tons of people looking to make transactions, fees go up or down based on people's willingness to pay to be included in the next block.
It also doesn't really increase the blocksize, but move some data outside of the blocksize calculation to make room for more transactions. The important difference is that Segwit is opt-in and depend on usage for it's effect, while a blocksize increase would immediately increase transaction throughput to its full capacity.
Nitpick here, but what will happen is that the cost of mining blocks will outweigh the revenue of selling its fees. This means that miners will at some point decide to stop mining bitcoin. If this happens abruptly, that would cripple the network for a while, since the difficulty adjustment function would not be able to adjust the difficulty in time. If the move is not sudden, but a progression over months/years/decades, that wouldn't cripple bitcoin.
If miners stop mining bitcoin, that doesn't immediately make bitcoin less secure. The hardware might be used to mine other chains, or could just be binned. If a single entity wanted to perform a 51% attack, she/he would need to buy up a lot of discarded hardware without notice.
A reduction of global hash power isn't a security issue per se; it is a multi-variable problem.
Over time, most bitcoiners foresee the transaction volume and demand rising, such that miner fees will compensate for the reduction in block subsidy.
So once we get to the end of new supply around 2140, the system will sustain from ongoing transaction fees.
Dan Held and I explore this in this interview if you're interested: https://stephanlivera.com/episode/81/