How exactly would personnel and funding be able to enforce a refugee cap? What is your vision there?
First, a cap should be determined. it could be 20K a year, 200K a year, 0.5M a year or 1M a year. i don't know what the capacity Europe could take but i am guessing it wont be able to cope with the current numbers. after that cap has been determined, only granting asylum to refugees that are in the camps and deporting the ones who illegally enter, will very likely greatly reduce the ones trying to enter physically. numbers would be a guess but since we are seeing virtually zero refugees trying to enter Saudi Arabia, Qatar and Oman, it is safe to estimate that what drive these refugees is not immediate danger but seeking better living conditions. if they wont be able to get those in Europe, they will not risk their lives trying to.
Even though the numbers would be reduced, some will still try and enter illegally, yes. the bulk of people who tries to do so via Greece and Italy would have to be stopped with extensive sea patrols and a new approach toward them. right now, the cost guards are intercepting the boats, unload the refugees in Europe and send the boats back. what they should do is destroy the boats they came with and deport the people back. this is probably the hardest and most expensive task, but it is manageable.
The land routes would need a physical barrier along with patrols and watch posts. The legal frame work for such a project could take something like a day of meetings if the European leaders were up to it. The construction of a physical barrier would take a few months since we are not talking about the wall of china, but a fence. such a fence already exist for a part on the turkey-Bulgaria border and the Turkey-Greece border has a river that serve as natural border for most of it. it took Saudi Arabia a year to build a fence 3 times the entire length of the Greek and Bulgarian border with Turkey so i can't see why it should take no more than a couple of months to finish and enforce such a fence on the entire Turkish border.
Even with all these measures, some will manage to enter, nobody doubt that, but it will be manageable and controllable for a large degree.
To say that it would take decades or require some kind of impossible legality is simply not true.