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The main controversy regarding the Altaic languages is that many of many of the earlier Altaicists, namely N.N. Poppe and especially R.A. Miller, have issues with their methodology and/or using unreliable data. Opponents of Altaic theory complained that the Altaicists were unable to come up with many cognates between the major Altaic families. But there are still plenty of Altaicists today, but most of them are Russian (which is not surprising since most of the Eurasian steppe has been Russian territory for the past few hundred years). A few of them do write in English, such as Alexander Vovin, whose article "Altaic, so far" (attached) provides a decent introduction and bibliography of the field today, as well as dispells some of the myths anti-Altaicists like to push. Anyway sites like http://altaica.ru/e_index.php contain tons of resources in both Russian and English.

Even if Altaic turns out unable to refer to a group of peoples and languages related by common descent, it is still a useful category. Even if they aren't related genetically they still share a good deal of similarities. For instance pretty much all of them practiced shamanistic/animistic religions, worshipped a patriarchal sky god and most were/still are nomadic. Even if those similarities are due to convergence, it's still a useful category.