Let's get real. The people who float across the channel on a dinghy are asylum seekers.
A fair correction. They will become migrants if successful. The reason I made the distinction though is because there are different rules for refugees fleeing war, compared to asylum seekers seeking economic opportunities.

There's no international law that says a country can't let people leave their country and go out to sea.
People trafficking falls into the category of modern slavery and is illegal under international law.

Current international treaties (general)Supplementary Convention on the Abolition of Slavery, entered into force in 1957
Protocol to Prevent, Suppress and Punish Trafficking in Persons, especially Women and Children
Protocol against the Smuggling of Migrants by Land, Sea and Air
Optional Protocol on the Sale of Children, Child Prostitution and Child Pornography
ILO Forced Labour Convention, 1930 (No. 29)
ILO Abolition of Forced Labour Convention, 1957 (No. 105)
ILO Minimum Age Convention, 1973 (No. 138)
ILO Worst Forms of Child Labour Convention, 1999 (No. 182)
Inter-American Convention on International Traffic in Minors
Nor does any country have an obligation to help another country's immigration system.
Countries are obligated to secure their borders with other countries. They are also obliged to take reasonable steps to ensure the safety of people in their territory. Letting people leave the coast for England on boats not fit for purpose is clearly failing in this obligation.

How can we get them to do cooperate? By being cunts over fishing?
By reminding them of their obligations under international law, and raising formal protests with relevant international courts.

Seems like it'd be hard to prosecute.
It's certainly not easy. That does not mean we should not use every resource at our disposal to attempt to bring these people to justice.

We could help the people who want to get here to get here without having to cross in a dinghy. That's one way.
Indeed, and we do. This is how legal immigration happens.

Let's assume the French are dicks or whatever and they dont' want to help us. What do we do then? Send a battleship?
We could treat our immigrants like utter shit to the point they want to leave the UK for France, and then do fuck all when they get on dodgy boats and head south. But naturally that would be highly immoral.

That's what France is doing.

No we shouldn't send a battleship to France. We should hit them with economic sanctions. Stop buying their wine.

The French can't tell asylum seekers they can't go on to the UK.
I'm not saying they should. They should simply uphold their obligations to protect the safety of people in their country.

Let's say a refugee comes from Syria. He travels first to Turkey, the border guard says "Who are you and where are you going?" The refugee says "I'm fleeing the war Syria and I have family in the UK I want to go there."
I'm glad you think that those trying to come here all have family here. I suppose this is why we have such opposing opinions on this matter. I believe they want to come to the UK because we treat migrants better than France does, we provide accommodation, money, education and health to migrants. Contrary to popular opinion, British people are particularly welcoming towards migrants, certainly more so than the French and Polish. Maybe not compared to Germany and Sweden, but we treat migrants very well, both legally and socially. That's why they want to come here. Not because they speak English, and not because they have family here (though obviously some do).