Ask Mr. Smarty Plants

All flowering plants are going to have fruits of some sort unless it is a plant that produces male and female flowers on separate plants and you choose a male plant.  Skunks, by the way, are omnivorous—eating insects, worms, small rodents and lizards as well as some berries and fruits. We can't guarantee that you won't get skunks in your yard, but here are some possibilities native to the San Antonio area that won't produce fruits particularly attractive to skunks:  

Ilex vomitoria (yaupon)

The male trees won't produce fruits.  The fruits on the female trees will be far enough from the ground that the skunks can't reach them.  They are generally eaten by birds before they would fall to the ground.

Leucophyllum frutescens (Texas barometer bush)

The fruits are small and not particularly attractive to skunks. It prefers full sun but will grow in partial shade.

Sophora secundiflora (Texas mountain-laurel)

The beans (fruits) of the mountain laurel are poisonous and not eaten by skunks or any other animals as far as I know.

Erythrina herbacea (coralbean) The beans of this plant are also toxic and would not be attractive to skunks.

Cornus drummondii (roughleaf dogwood) Fruits more likely will be eaten by birds before they fall to the ground.

You can go to the Recommended Species page and select South Texas from the map or pulldown menu and use the NARROW YOUR SEARCH option in the side bar to select 'Shrub' or 'Tree' from the GENERAL APPEARANCE category and 'Partial Shade' from the LIGHT REQUIREMENT category to find more recommendations.

Ilex vomitoria

Leucophyllum frutescens

Sophora secundiflora

Erythrina herbacea

Cornus drummondii