How does Speed Booster® affect the depth-of-field?

The short answer is Speed Booster® gives essentially the same depth-of-field effect as if a full-frame camera body were used.

The long answer is complicated. If we are referring to depth-of-field in the mathematical sense, that depends on the aperture, magnification and circle of confusion (CoC). Magnification in turn depends on distance and focal length. The 50mm lens now becomes a 35mm lens which behaves very differently in terms of perspective. The question is, do we still keep the distance the same? Should the CoC be kept the same? There are many missing variables we need to choose and fill-in before we could get a meaningful answer. When people claim Speed Booster® does not change the depth-of-field, they usually neglect to state the implicit assumption that the distance is kept the same (thereby changing the object size) and the CoC is kept the same. The same logic would lead to the conclusion that an APS-C camera has the same depth-of-field as a full-frame camera, too, which under the same implicit assumptions is mathematically true (the depth-of-field formula is format-size-agnostic, after all), but with which many people would disagree from practical experience.

However, when most people ask about depth-of-field, they are not interested in mathematics, but rather, they are after a certain kind of shallow depth-of-field "look". If this is the case, the short answer above applies.