A certain amount of job training happens through the basic experience of showing up for work everyday. But in some cases, more specific training is called for, which can either be sponsored or provided by an employer, or by the worker. Here's some evidence from the recently released 2015 Economic Report of the President, by the Council of Economic Advisers, showing a decline in employer-provided and on-the-job training in recent decades. This data is collected only irregularly, when the Census Bureau decides to include the "module" that includes this particular set of questions. But with the most recent data in 2008, it seems pretty unlikely that employer-provided and on-the-job training have risen in the aftermath of the Great Recession.
Thus, types of skills, who pays for them, and how they are paid for can be sliced various ways. But just looking at the overall pattern, a decline in employer-sponsored and on-the-job training suggest that workers who wish to keep building their skills are getting less support from their employers.