If you are planning to for Everest Base Camp Trek and Gokyo Lake EBC Kalapathar Trek, you preferably choose the Pre-Monsoon season through March, April and May or in the Post-Monsoon season from late September, October, November.
Being at such a high altitude, the weather in Everest Base camp is never warm. The weather fluctuates drastically and dramatically. You will experience sudden precipitation, snowfall, or changes in temperature. However, there is some particular time of the year when the weather is more favorable than others.
Springs and autumns are indeed the best times to go for any trek or hikes. March, April, May, October, and November are the best months to travel to Everest Base Camp, and the reasons are relatively simple. There is little-to-no rainfall, the temperature is favourable, the views are spectacular, and the chance of random snowfall is also very little. But it's the best time of the year for this trek. The camp is likely to be crowded and at its busiest phase.
If you're someone who hates crowded camps and jolting for space with fellow trekkers on the way, winters are the perfect time for you as it is relatively less crowded, and you will enjoy the views and space mostly for yourself. But since it's winter, and that too in the Himalayas, be prepared for extreme weather conditions. It is going to be freezing without a doubt. Make sure you pack as many warm clothes as possible.
One great thing about going to Everest Base Camp during the winter is that the skies are clear and offer the best mountain views.
One season where you should not consider the Everest Base Camp trek is the monsoon. The trails are going to be muddy and wet, and you could fall and injure yourself. Don't mind yourself always getting wet even while wearing your raincoat all along the way.