The simplest way to measure alcohol dehydrogenase activity is by the absorbance increase at 340 nm when NAD+ is converted to NADH. This may be difficult to do in a complex extract because there may be many other things that absorb at 340 nm, and because the extract may be turbid.
An improvement on the simple method is to use a coupled enzyme assay that produces a color at a longer wavelength and also increases the sensitivity of the measurement. For example, you can add the enzyme diaphorase, which is commercially available, to couple NADH production to reduction of the chromogenic substrate XTT. This shifts the absorbance to 490 nm, I think, and is several times more sensitive than the 340 nm absorbance of NADH. There are other commercial chromogenic and fluorogenic methods available as well, such as those using resazurin.