Caffein, which is the main active ingredient in coffee is known to be dehydrating. However there is no conclusive scientific evidence that caffein intake in normal dosages has detrimental effects on the quality of your voice.
One study by Akthar et al. (1999) showed that the ingestion of 250mg of caffein, which corresponds to 2 to 3 cups of coffee, produced alteration in the voice of 8 individuals. I would take this result with a grain of salt though, since a study performed with only 8 subjects, without control group or anything, can hardly be called representative or conclusive in any way. Also they say that the effect produced varied substantially between individuals.
Another a study by Franca et al. (2013) is showing that the ingestion of 100mg of caffein shortly before using your voice does not have any negative effect on the quality of your voice. I think this is a more plausible study design, 100mg corresponds to a cup of coffee and the experiment is controlled with one group taking placebos.
Finally a recent review (Alves et al. 2017) of the literature on dehydration and vocal quality found that dehydration is, surprise, bad for your voice. They also found however that "systematic hydration is the easiest [...] solution to improve voice quality".
Thus I would conclude, that as long as you take care to stay hydrated, meaning drinking water with your coffee (which everyone should do in any case) moderate amounts of coffee consumption will likely not have any detrimental effect on your voice quality. The evidence for that seems to be mainly anecdotal and not scientific. If you struggle with some vocal chord disease for example or you are in recovery of one, maybe coffee could be ill-advised. Akthar et al. (1999) hint to something like that. There's no proof that it's damaging in those cases either, but just for the sake of being overly careful rather than ending up doing damage (anyways it would likely be only temporary) to your voice, if you are professionally dependent on it, maybe restrain from coffee consumption in cases of diseases etc.
References:
Akhtar, S., Wood, G., Rubin, J., O'Flynn, P., & Ratcliffe, P. (1999). Effect of caffeine on the vocal folds: A pilot study. The Journal of Laryngology & Otology, 113(4), 341-345. doi:10.1017/S0022215100143920
Alves, M., Krüger, E., Pillay, B., Van Lierde, K., & van der Linde, J. (2017). The Effect of Hydration on Voice Quality in Adults: A Systematic Review. Journal of Voice.
Franca, M. C., Simpson, K. O., & Schuette, A. (2013). Effects of caffeine on vocal acoustic and aerodynamic measures of adult females. In CoDAS (Vol. 25, No. 3, pp. 250-255). Sociedade Brasileira de Fonoaudiologia.