I've had a beautiful genoa fig tree for several years and it produces an abundance of figs.  However, I've only had a few of these delicious figs over the years because most drop before ripening.  What do I need to know?

There are a lot if factors.

Overwatering
Underwatering
Poor plant heath
Maybe it isn't a Genoa.
Maybe is is a smyrna type, getting minimal pollination.
Climate is too hot.
Climate is too cool.
...

How do I figure out what is going on?

A good start:

Where do you live, regionally (or city/state)?

Where did you acquire the tree?

What makes you say it is a Genoa?

Is it planted in-ground or in pots?

What is your general care regimen for fertilizing/watering/etc.?

Where is the plant, location-wise (north side of house, out in the open, full sun, full shade, etc.)?

I've got a similar problem with my LSU Gold.  Last year, it produced quite a few figs.  They were gorgeous!  The tree is 4 years old and in a 30gal container.  A month ago, I planted several of my trees in the ground.  The first was my LSU fig trees.  All the other trees had started to leaf out except my LSU Gold.  I took a chance, root pruned it and placed in its new spot.  

I'm in the south San Diego, CA area.  Purchased the tree at Home Depot and it had a "genoa fig" tag on it.  Planted it directly into the ground.  I've had the tree about 8 or 9 years and haven't quite figured out what to do.  Last year I mulched with grass clippings and tried to do deep watering on a somewhat regular basis, thinking maybe it wasn't getting enough water, but it still dropped fruit.  The tree is on the south side of my home and gets quite a bit of sun  The soil is thick and has lots of clay but it's really dry, but I've had a plum tree next to it that produces wonderful plums.

So glad to hear that figs can be a bit of a mystery plant with the ability to drive one crazy!

I wonder if it's actually a Celeste, based on the fruit dropping?  ;)