Well, there is a reason it's called the "rain forest".
This isn't southern California, all that lush vegeatation is the result of lots and lots of rain.Now, you might get lucky and arrive during a relatively rain-free stretch of days. A lot depends on what else is going on in the region weather-wise. For example, a tropical system might wander in and park itself off of Honduras-Nicaragua for a while, and San Jose gets a week of torrential downpours. Or you get week of constant sun. It's hard to predict exactly what will happen, but the typical pattern is for rain to develop in the afternoon (usually hard but brief downpours) then to clear out by evening.
One solution is to go whitewater rafting. You're gonna get wet anyway, so a little rain won't matter!