Technically you don't need to be online 24/7, just often enough to detect the fraud before the timeout, default is two weeks I think. Or you can pay someone a tiny fee to monitor the blockchain for someone publishing an old state for you.
In practice, nobody is trying to commit fraud, because : a) if you get caught you get punished severely b) most channels do not have enough BTC in them
Lightning is for small, instant, fast payments, average channel size is about 0.02 BTC.
The bigger problem is how to maintain up to date backups, and not accidentally publish an old state when recovering from a crash. We're working on it. Taproot bitcoin upgrade brings some interesting improvements we can build upon to improve things further.
EDIT: forgot to mention this is only really a problem if you want to run your own non custodial solution ( which I do). For most people, using a 3rd party custodial solution for their low balance hot wallet seems to be the norm.
A.k.a trust a third party.
> For most people, using a 3rd party custodial solution for their low balance hot wallet seems to be the norm.
Just like Venmo or PayPal?
*Bitcoin Cash enters the chat*
True, but in a very limited sense. You're trusting a third party to protect you from your counterparty posting an out of date state. And adding further complexity, your counterparty gets punished if they get caught publishing this state, and they have no way of knowing if you are watching or not.
So, from a game theory perspective, it is unlikely for this to even come into play.
Oh, and you can technically have multiple third parties doing this.