The ingredients in beer are not unkosher. There is nothing in craft beer (water, hops, barley, yeast) that is not inherently Kosher.
However, that doesn't make beer made with these ingredients Kosher. For instance, they could have been made in the same tanks as some other non-Kosher product and the tanks may not be properly cleaned in between, or perhaps some of the ingredients were in touch with non-Kosher items before getting to the brewery.
Two other esoteric items would make them non-Kosher. One is how the yeast is grown. Redhook, the first major brand to be Kosher certified, had to change the medium in which they grew the yeast. Another is the finings that may be used to clarify the beers. Some of the fish finings come from unkosher fish which would make the beer unkosher.
Lastly, in a bit of real trivia, for the super-Orthodox, including the chief Askenazi rabbi in Israel, beer has to be made with 'old barley' which is barley that sprouted before the second day of Passover. All Sam Adams beers shipped into Israel are made from old barley.
The bottom line is that it is not Kosher unless it is certified by a Kosher certifying agency. They usually do a pretty thorough check of the purchases, brewery etc... and spot visits one or two times a year. There are a large variety of logos of these agencies and the logo is stamped on the bottle label.
Coors/Molson is/are Kosher, Redhook is Kosher, Sam Adams is Kosher. A host of others too numerous to mention includes a lovely small brewpub in Oregon that doesn't charge Jewish customers for beers on Saturdays because the owner knows they are not supposed to carry money on their Sabbath.
All that aside, Kosher for Passover is a contradiction in terms if you believe beer is a fermented grain, which is our usual definition. Since fermented grains are not allowed on Passover, beer cannot be Kosher For Passover
—Steve Frank is one-half of the Brews Brothers. With Arnold Meltzer, their work has appeared in the American Brewer, Mid-Atlantic Brewing News, and other periodicals.
Mr. Frank's answer to the question of whether or not beer is Kosher (and Kosher for Passover) was originally posted on DC-Beer, an on-line group of good beer partisans in the Washington D.C. and Baltimore area. For a more detailed exegesis on Kosher beer, click on Divine Beer, a pdf of their article originally published in American Brewer Magazine. Reprinted here with their permission.