You may just have a clogged downspout, insufficient drop to downspout, or not enough downspouts. I don't know all the facts of your problem and what you consider a "good rain", but I would look to see if something is preventing the gutter and downspouts from functioning properly. They may overflow during a downpoar but shouldn't otherwise.
If you can correct problems with the current water removal system, then I would start there. Not too expensive either. You could test the system with a garden hose, unless the gutter is at a height where you are not comfortable working, then you could hire someone.
I don't know if anyone has any good tips for unclogging a downspout? Sometimes, a second downspout is necessary at the other side of the run.
Let us know what you find. Once you are sure the water removal system is working at peak effincency, then I would look into diverting the ground water. Or, I could just be "all wet" with my opinion.