Hi Brittany -- that is a very interesting question! The solar cycle has never stopped completely, but there have been extended periods of very low solar activity. One example of this is the Maunder minimum, between 1645 and 1715, when very few sunspots were seen. (For comparison, a "usual" solar cycle will have a period of lots of sunspots approximately every 11 years, which is "solar maximum". ) This time period coincided with the "Little Ice Age" where rivers froze in some parts of the world that didn't usually, and snow fields didn't melt as they usually did... So it is certainly possible that if the solar cycle stopped, climate on Earth would be affected. Diminished solar activity means a decrease in the sun's ultraviolet radiation which impacts ozone formation in the Earth's stratosphere. Jet stream winds may also be impacted by the the less active, and thus dimmer Sun.
However, climate can be very complicated, and it is possible that volcanos also played a role in the cooling period that includes the Little Ice Age -- see
There's still a lot of work to be done in figuring out what happened during the Maunder minimum, and more generally, what effect the solar cycle has on the Earth's climate.