How do airlines make sure that pilots remain skilled enough to operate aircraft? What quality assurance is there for certified pilots?
Put quite simply for this answer - The Licence a pilot holds is the assurance that the pilot has received the appropriate level of training.

Each country has it's own standards for training, their own requirements, and their own limitations on the operation of that licence. In saying that, MOST countries standards are reasonably in-line in regards to Aeronautical Experience required to sit each flight test.

In Australia, the requirement to sit a Commercial Pilot Licence is a minimum of 200 hours Aeronautical Experience (150 hours if ALL training has been conducted on an Integrated Course, now to be recognized under Part 141). From there, there is a break down of what that 200/150 hours must consist of. 

Now, having a Commercial Pilot Licence (again, speaking from an Australian point of view) does not give a pilot the right to sit in the left seat of an Airliner. You need further training such as;


Multi-Engine Command Instrument Rating
Type Endorsement (for the intended aircraft)
Crew Resource Management Training
Dangerous Goods Training
Senior First Aid (Not mandatory, but common)
Company, Emergency, Cyclic and Systems checks (per company)
Air Transport Pilot Licence Theory Exams (in some cases, Airlines may not even look at you if you haven't completed these - In
Australia, it is an unwritten must-have!)


And thats just to sit in as a Co-Pilot.. 

Captains need years of experience, and a lot more training and check to get into that coveted Left Seat.

So again, to bluntly answer your question, regardless of country, there is law in place that stipulates that Pilots must be appropriately trained to fly airliners.