Silicone caulk in the gaps will be the ticket. Preferable if you use an acetic acid based caulk. (read the ingredients)
Humidity cures the caulk, so smooth it with a dry finger. I've made amazing messes with the old wet finger trick until I learned that. Wet finger works for smoothing latex caulk.
If you use the acetic acid based caulk you only need to wait for it to skin over. Otherwise, let it cure for 24 hours.