"Do the media..." or "Does the media"?
Solution 1:
media, like audience, crowd, committee, herd, public and family, is a collective noun and is addressed as such. It is understood that media is made up of not only many news sources, but many types (television, radio, movies, etc.) as well. The singular component is called a medium.
There are, further, so called collective nouns, which are singular when we think of them as groups and plural when we think of the individuals acting within the whole (which happens sometimes, but not often).
David Letterman apologizes to his wife through the medium of television.
We do not say, do the public approve of these restrictions? (AmE, some difference in BrE and others.)
Even so, sometimes media is treated as a plural, not collective, noun, so we do see examples such as
Do the media fairly explore issues, or do they impose their own positions? The influence of the media is increased by the fact that campaigns today have ...
This is probably done often enough (BrE and others) to be seen as a style issue, hence the confusion (esp. to AmE speakers/readers.)
edited to add:
Solution 2:
Like other collective nouns, it may take a singular or plural verb depending on the intended meaning. If the point is to emphasize the multifaceted nature of the press, a plural verb may be more appropriate:
The media have covered the trial in a variety of formats.
Frequently, however, media stands as a singular noun for the aggregate of journalists and broadcasters:
The media has not shown much interest in covering the trial.
(The American Heritage® Dictionary of the English Language, Fourth Edition copyright ©2000 by Houghton Mifflin Company. Updated in 2009.)