I'm just starting to brew and I was wondering if my closet would be cool enough to properly ferment. During the winter my house is usually in the mid 60's. Should I still do something else to regulate the temperature?
It depends on the yeast you're using with your brew. If you're using an ale yeast that is best in the low-to-mid 70's, something like a Brew Belt is a relatively cheap and easy way to keep the temperature slightly higher than the ambient temperature of your closet.
If you're brewing with a yeast that's best in the upper 60's, you may be fine with that temperature in your closet.