Hi Carol,
June is very early in the rainy season so anywhere you visit shouldn’t be that bad. First off, we’d recommend taking a look at our blog post about visiting Costa Rica during the rainy season: https://www.twoweeksincostarica.com/visit-costa-rica-rainy-season/. The box at the bottom of the post has info about what the weather is like in the different regions. Basically, the post says that Guanacaste Province (NW Costa Rica) is the driest so will be the most likely to not have much rain. Another beach town you should look at is Manuel Antonio on the Central Pacific Coast (more info here). It has more rainforest so receives more rain but is still really nice in June and is particularly good for families. Have a great trip!