They can't apply for asylum in the USA until they are physically in the USA.
Sure they can. I've just read an article that makes it clear people can apply for asylum for the UK from outside of the UK.

If they chose to go to the USA, that's not the same as us "sending" them.
By "send" I mean "transport". No need to be nitpicky.

What if there's a large number of them trying to do it all the time? What if they're setting off in the middle of the night? Should we turn our beaches into an armed camp to stop them, or is it enough that we just stop the ones we can catch?
What do you think we should do? Fuck all? Just leave them to it? Have a nice cup of tea and watch?

Is patrolling the waters going to stop 100% of them from getting out to sea?
No.

And btw, do you think France just emptied the English channel of any French ships? Or do you think that they patrol it, but if they see a dinghy crossing they wave it through?
The Channel is one of the busiest shipping lanes in the world. It's not the Atlantic Ocean. They should monitor their coats.

Can they stop everything? No. But they should be doing everything they can to stop as many as possible.