The south-west monsoon is still active through June, affecting the south-west quarter of the country but it’s already blowing itself out and generally consists of short, sharp showers, so you shouldn’t be put off from visiting such areas as Galle or the tea country.
In fact as long as you don’t mind a little rain June is a very comfortable time to visit, with average temperatures in the high 20°C (except hill country) and fewer visitors meaning more attractive prices. The east coast remains the place to head for the beach, although the west and south coast beaches are a viable alternative.