Ask Mr. Smarty Plants

Parthenocissus quinquefolia (Virginia creeper) will climb on anything in its general vicinity, but it is not as pervasive or invasive as, say,  Vitis mustangensis (mustang grape). Generally, it won't take over completely like the mustang grape, but it will climb the trees. Virginia creeper's potential for harm to the trees is that it might become so dense that it shades out the leaves of the tree and interferes with the tree's photosynthesis. So, all in all, it might be better to avoid putting Virginia creeper down as a ground cover, or at least be prepared to monitor it and keep it out of the trees.

All in all, grasses are the best solution to erosion control because of the dense fibrous roots that they produce. It sounds as if your area does have at least some shade so here are a few grasses that do well in partial shade that have attractive features.  My guess is that you don't want something that is going to require a lot of mowing and these grasses are bunch grasses that are attractive both in their green growing stage and even after they have matured and "gone to seed". The old growth can be cut back once the new growth begins in the spring.

Schizachyrium scoparium (little bluestem)

Chasmanthium latifolium (Inland sea oats)

Sedges are another possible ground cover that don't require mowing and stay green all year long. Here are three possibilities:

Carex blanda (eastern woodland sedge)

Carex planostachys (cedar sedge)

Carex texensis (Texas sedge)

Finally, here are a couple of low-growing herbaceous plants that do well as groundcovers alone or mixed in with the grasses and/or sedges.

Calyptocarpus vialis (straggler daisy)

Phyla nodiflora (Texas frogfruit)