Let’s ask the question the other way round, what could make wine not kosher. Basically there are three points that need addressing:
- Jewish agricultural laws
- Wine ingredients
- Handling by non-Jews
Jewish agricultural laws here refers to the biblically mandated laws of orla (you cannot use the fruits of the vine in the first three years after planting), shmita (you cannot actively cause growth for any plant in every 7th year) and truma and ma’aser (two types of tithes). Besides orla, the laws only apply to the land of Israel. The excellent Kosher Wine 101 by winemusings gives you more details about them. My understanding is, that orla only applies to Jews. So if I were to buy a wine from a non-Jew outside of Israel, these laws would be non-issues. In Israel, wines that have a hechsher can be trusted to adhere to all these laws. Wines without hechsher from Israel may pose a problem though.
Let me skip to the third point, handling of wine by non-Jews, before discussing the second. The problem is, if at any point in the winemaking process, a non-Jew (or according to some a non-observant Jew) touches the wine itself or an open container. This rule was introduced because of fear that the non-Jews would use the wine for idol worship (Avoda Zara 29?). As today’s wine is made in factories and nobody would dream of using it for idol worship before it goes on to the supermarket shelves, the conservative movement has argued that this point is irrelevant for the wine produced in the Western world (Elliot Dorff: On the Use of All Wines). Another possibility to get around this point in some way is to buy "mevushal&qout; wine, wine that has been flash-pasteurized. Although I think the only wines that are mevushal are those with a hechsher, because why on earth would you do that to your wine otherwise?
Last but not least, the problem of ingredients. Wine, as any processed food nowadays, can have many ingredients besides grapes and some of them pose kashrut problems. Beef blood may be forbidden in some countries including the US and Europe, but this still leaves other potentially problematic ingredients like casein (which would make the wine dairy), acid blends, or gelatin (winemusings: Kosher Wine 101). The article lists also yeast as a concern for Pessach, but it would have to be yeast from the 5 grains to pose a problem, I am not sure if the article is indeed saying this is what is used. And of course the ubiquitous kitniot: corn syrup (only in the US I suppose). Not all of the used ingredients may appear on the label (e.g., fining agents do not remain in the bottle and therefore do not have to be declared).
Of course, buying only wine with a hechsher solves (not only) the problem with ingredients. But what about someone who holds that it is not a problem if a non-Jew touches the wine, but still wants to avoid non-kosher ingredients? Or does not consider a particular ingredient to be a kashrut problem (e.g., gelatin). I am not aware of any conservative hechsher that would address this need. Wikipedia claims that there are "unfiltered" wines, so this might be a partial solution (Wikipedia:Vegetarianism and wine). An alternative route is to check Barnivore, the "vegan beer, wine and liquor guide" or a similar page (if you find one more geared towards Europe let me know!) to check the ingredients. Couldn’t we crowdsource kashrut lists that way, too?
Bottom-line: Besides the hotly discussed issue of non-Jews touching the wine, there are other problems with wine that are worth looking into.