As mentioned in another post, there are mosquitos all over the tropics, which means all over the Caribbean. The mosquitos are not unique to Culebra, nor, would I consider them to be a "problem" in Culebra.
I did a triathlon in Dorado in the last year. It is not "close" to Old San Juan, but rather about an hour or so.
While the beaches on the big island (including Dorado) are pretty, you don't have the powdery white sand & turquoise waters like in Culebra.
Even if you chose to do a "day trip" to Culebra (which I actually don't recommend as Culebra has so much to offer & with the time/energy to get to /from the outer islands a min. 2 day visit is always best) you will still encounter mosquitos. Although, you will meet mosquitos on the main island as well.
Palmetto GH, Culebra