Before we entered the EU there was a common travel area with the Republic of Ireland, and after we leave there will also be a common travel area. The checks at the border for people will be the same before and after exit. There is substantial co-operation across the border over criminals today and this will remain.
After we leave the EU it will still be possible for a person from another EU member state to get into the Republic, subject to EU controls on criminals and illegal migrants, and from there to cross into the UK in Northern Ireland. The UK system of control over migrant numbers will be exercised for the whole UK by the need for a work permit if someone wishes to get a job, by the need to establish entitlement if they want to receive a benefit, and the necessity to prove they are legal immigrants if they wish to open a bank account, rent or buy a property or get a car licence.
It is difficult to see therefore why an illegal migrant from the EU would bother to go through the tortuous journey via Dublin, only to find on arrival in the UK that their illegal status made it impossible to live a normal life or benefit from the good things that brought them to the UK.