Most (non-western) countries do pretty well by making it clear that those who enter illegally will be arrested, charged, imprisoned, and then deported (at their own cost) rather than given free benefits.
Believe it or not, the UK already does this to people for whom there is no plausible asylum claim. The problem, if you want to think about it as such, is that the UK currently believes in treating potential asylum claimants as such rather than automatically labeling them as criminals.
Bad news, budget cuts mean we can afford a flounder with a paper fan. His name's Steven and he lives about twenty yards east of Lulworth Cover, but we reckon he's okay to commute to work.