Suppose workers differ in quality, where "quality" is defined as "quality in the eyes of potential employers". In equilibrium there will be an equilibrium relationship between the distribution of wages and quality. Higher quality workers will get higher wages.
Start in equilibrium. Now assume a fall in demand for all types of labour. If wages are flexible, the whole curve of wages against quality will shift down, with no obvious effect on unemployment, or on relative unemployment rates for different qualities of labour.
But if wages are sticky, what happens? Firms will cut employment, and some workers will become unemployed. There will be an excess supply of labour for all jobs. If any employer does have a job opening (and some will) he will hire the highest quality worker in the queue. So higher quality workers will tend to get re-hired (though at lower wages than they earned previously, and working alongside lower quality workers). And lower quality workers will tend to remain unemployed.
In a recession, a bigger increase in unemployment rates for lower quality workers is exactly what we would expect from sticky wages. It's an observation that confirms the theory of sticky wages.