"If the authorities came across an overladen boat half awash and in danger of sinking what should they do?"
Firstly, do not make too much of an effort to "come across" them.
Secondly, if they do, let nature take its course. Sorry to be harsh, but if the European nations continually pluck these people out of the drink and go on to feed, clothe and house them the problem will not only persist but, as has been adequately demonstrated, continue to worsen.
As I have said before the numbers involved here are such that southern Europe (and, courtesy of the ridiculous Schengen agreement, all the rest of the continent) is under invasion, nothing less. A hundred thousand people have landed on that tiny island so far this year. Many of them have had their boats helped into the harbour by the Italian navy. The Italian authorities (some of whom are "proud" to see the area as a haven for migrants) then assist their passage into France and on to that country's northern ports.
The boats should be taken into tow and returned to north Africa. Unless some really serious action is taken to stem the flow Europe might as well charter a few ships and bring them here in a little luxury.