I think that most phototransistors will have useful response in the visible red-LED wavelength. If you have one to hand, it won't take a few minutes to try out.
The idea of a phototransistor is that the base junction works like a tiny solar cell and generates the base current to turn the transistor on. A three-lead unit brings the base out so that you can supply a bias to adjust the sensitivity. For most binary on/off applications, I don't think you'd need that. If you were analogue sensing light levels, then you'd maybe want to bias the transistor into a more linear range for your particular light level, or something more sophisticated.