The monsoon usually reaches Nepal in early June, and certainly by mid-month. Overall rainfall for June is lower than for July or August, but that's mainly because the monsoon hasn't yet hit in the first week or two of the month. Rain clouds and humidity build to uncomfortable levels in Kathmandu, and while temperatures are much cooler in the mountains, conditions aren't optimal for mountain viewing.
But, even once the rains have started, it generally doesn't rain all day every day, so if you're prepared for wet conditions and understand that humidity and rain clouds may obscure your views, June isn't an awful time to trek in the Everest Region. Plus, the higher you go in altitude, the less likely the chance of rain.
Crowds & Costs
Aside from expeditions returning from Everest, there aren't very many trekkers on the trails in the Everest region in June. You're unlikely to struggle to get a bed in a decent teahouse along the way, but some may close for the monsoon season. Always trek with a guide—one of the benefits is that they'll know which of the best places are open in the low-season.
If you start trekking in late May or early June, you may get lucky and not encounter much or any rain while trekking in the Everest region. However, the building humidity is likely to obscure the best views. If trekking later in the month, you will certainly encounter some rain. As treks in the Everest region normally rely on access by air via Lukla rather than by road, the chances of encountering problematic landslides are minimized, but most people would agree that trekking in the rain isn't the most fun experience.
If you have your heart set on trekking all the way to Everest Base Camp (EBC), you'll probably find that it rains most at lower elevations (although nowhere in the Everest region is exactly 'low'), and that the higher you go, the less rain there will be. So, if you're willing to brave a few days of trekking in the rain, you may still be able to check out EBC itself in decent conditions.
A good way to avoid long days trekking in wet conditions in the Everest region is to base yourself in Namche Bazaar and go for day hikes from there, and the odd overnight hike if conditions are favorable. Namche Bazaar is two days' walk from Lukla. While you won't make it all the way to Everest Base Camp, you'll see and experience some beautiful things, and there's the chance you'll catch some early-morning or post-rain views not far from Namche Bazaar.
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What to Bring
It's important to have good waterproofs for trekking in June, and enough changes of clothes that you don't risk having to re-wear the previous day's wet clothes. Many trekkers find an umbrella to be more convenient and comfortable for showers, though if it looks as though the rain is setting in for a while, a comfortable raincoat and even rain pants are a good idea.
Check out this handy guide to packing for a trek in the Himalaya: Trekking Gear List for Nepal.
Getting There & Away
The quickest and easiest way of getting to the Everest region in any month, including June, is to fly from Kathmandu to Lukla. However, these flights are often delayed or canceled because of the weather or conditions in Kathmandu, Lukla, or en route. The chances are even higher in June, because of the wetter-than-normal conditions. If flying to/from Lukla, leave yourself plenty of time to make important international connections, and take a good book or pack of cards.