Why would there be a need to put a stop to it? Especially if, as you suggest, it is impossible? Also, which lefty labour did you have in mind? An hour or so ago I read that Starmer is a traitor to the left, betraying Corbyn's committed opposition to all things Israel....I'd be interested to know what your views and opinions are with how the lefty Labour Party would deal with this issue. If anyone thinks they can put a stop to it then they're deluded.
Labour will have their work cut out, to be sure. The choice is do nothing or throw money at it. Voters (especially those obsessed by 'people coming over here') do not want to spend a single penny on this. That is why we spend little processing these people, and simply farm them to temporary accommodation (from which they abscond - my brothers; job is to find temporary accommodation for asylum seekers and this is what he tells me is happening - all due to a refusal of HMG to spend the necessary money).
So if Labour decide to do something they will have to spend money. Then the right will boo. BOO!
My view is that to 'stop the boats', this has to be done in France, from whence the boast emanate. To do this requires work done on the French coast, with lots of patrols, supplemented with lost of drones patrolling coasts where boats can feasibly leave. This requires co-operation with the French, and the spending of lots of British money. But it isn't impossible. In effect you 'shoo' the boats back to France before they have had a chance to enter 'neutral' or 'British' water.
There, that took me ten minutes to think up. I am sure Labour can do better when they get in. If I were them, though, I would offer no solutions publicly yet. Sunak is now so desperate he may simply steal them all.
Beyond all this more work should be done rebuilding relations with the French and the EU. How do the 'illegal' immigrants get into France in the first place? Where are they sheltering before they get on an 'illegal boat'? I get the impression France no longer bothers spending money to stop boats leaving. Certainly if Braverman were French president she would suggest it is 'not our problem'. This is all the toxic product of years of sabre rattling Brexiteering. We can't turn back that clock, but Labour must work with France and the EU if a solution is to be found.
So collaboration with France and spending money. If this halves the number of boats in a year, this will still be seized on as Failure by the right wing press, and not only that, a Waste of Money, so whatever labour do they will lose votes as a consequence. This is because the tories are not bothered about stopping the boats. This is because they have successfully weaponized 'illegal' immigration in their true war, the war against Labour. Judging by your attitude, the tories are winning this war.
Aside from all that, this should not detract from the fact that the current tories are utterly useless, some of them vile scum, and they need to be voted out.