They're not arriving illegally. It's perfectly legal to turn up at a country and claim asylum. It's illegal to stay if their claim is rejected This is completely wrong.
Anyone who bypasses immigration control and enters the UK clandestinely or with fake passports etc is an illegal entrant. It doesn’t matter why they’ve come. They have committed an offence under immigration laws and it’s an offence which is technically punishable by fines and/or a custodial sentence (link is to the .Gov website which explains the exact laws) https://assets.publishing.service.gov.uk/media/6650731b8f4cb8fef9f64fad/Irregular+or+unlawful+entry+and+arrival.pdf#page6
However, it’s recognised that this is one of the few ways for someone to claim asylum in the UK, so nobody is prosecuted. The vast majority are dealt with administratively, which means that they’re given the opportunity to explain why they’ve come to the UK. Most will claim asylum and then become what is commonly known as asylum seekers. If their application is granted, they become refugees. If it’s refused and they don’t qualify for leave under some other category (eg, Humanitarian Protection, Article 8 etc) they then become removable from the UK. They aren’t breaking any laws or committing any offences by staying if their claim is rejected. They sometimes have a right of appeal against the decision to refuse their claim and they’re perfectly entitled to remain here while that’s ongoing. If they don’t have a right of appeal, or their appeals have all been refused, then they can choose to leave voluntarily or be removed by the Home Office.
Also, just to repeat what a couple of very sensible posters have already said, Brexit made sod all difference to the small boats. Some of you seem to be under the impression that we were returning thousands of people each year. We weren’t, and we accepted more people under this agreement than we ever removed. The stats for this are published so feel free to check if you don’t believe it.
Lastly, processing claims abroad so we can provide safe passage to the UK and stop fatalities is not going to stop the small boats. Anyone who thinks it will, clearly doesn’t understand how migration works. If you’ve come to the UK from France, or Belgium or any other EU country, then you clearly want to be in the UK over any other safe country. We accept the vast majority of asylum applicants but we’re still refusing thousands each year. If these people are so desperate to risk their lives in a dinghy to get here, do you think that being told ‘no, sorry but you can’t come’ is going to stop them? Of course it isn’t. Except now, they know that we’ll have a record of them, so will be keener than ever to keep their heads down and avoid the authorities. Which leads to greater risks. Which leads to more fatalities. Which nobody wants.