Does a too low fermentation temperature affect taste?

Each yeast you use will have a recommended temperature range, however, that doesn't mean it's ideal to swing wildly, even within that range. You can typically get away with temps below the recommended range, but above the range will definitely produce off flavors, as will stressing the yeast out with wide temperature swings over the fermentation period. Even submerging your fermentation vessel in water is helpful if you don't have tight temperature control. Finding a room or a fridge you can dial in to be a constant temperature (and one recommended for your yeast) will improve your fermentation characteristics.

I use ale yeasts at fairly low temps all the time and I haven't experienced any so far that have negative characteristics at 60F. Some, like WY1007, 1728, or 1056 even work fine down into the 50s. Yes, a lower temp will produce fewer esters, but in general that's what I'm looking for. Whether fewer esters are a positive or negative effect depends on what you want.

As you reduce the temperature you need to compensate with more yeast, just as you do when brewing a lager. E.g. for a 1.050 ale fermented at 60°F a 2 liter starter would be the minimum.

Alternatively, if pitching from a smaller starter, increase the temperature slowly after primary fermentation is almost complete - e.g. 3 days or when you hit 75% of the target gravity - raising the temperature then will not produce significantly more esters and will increase the yeast metabolism so they continue fermenting rather than drop out.