Re: Stupid question here, Cross-training shoes
Yes and no.
I wear Nike Free 7.0 trainers, and these are GREAT shoes. They develop some of the foot muscles you normally wouldn't, and I have found this to greatly help my balance and footwork. They are very light shoes, and have excellent foot support. I've played basketball in these shoes as well, and they are great for basketball but have almost no ankle support so injury would be quite likely.
The biggest problem with using trainers for tennis is outsole wear. Anyone who plays hard knows that shoes wear out pretty quickly and the expense can add up when you're getting good shoes. Some trainers might have outsoles that mark up the court as well, so be careful with what you chose.
Lateral support is paramount in a pair of tennis shoes, second to me would be how it wraps your foot, and third would be outsole wear. If you can find trainers that are good for these three things, then you might have a winner. Basketball requires more ankle support, and more cushioning because of the jumping. You can either get a shoe for both tennis and basketball, and end up buying shoes more often, or save up to shoes for each sport.