TL;DR The aperture stays the same, independent of the sensor size, for both exposure as well as depth-of field purposes.

The aperture has two effects, which have to be considered individually:

• exposure
• depth of field (DOF)

For exposure, an f/1.4 is an f/1.4, no matter how large or small the sensor area in the camera body. (Effectively, the "crop sensor" is the same as taking a full-frame picture and cropping afterwards, and you need not adjust brightness/contrast when cropping in your favorite image editor, do you?)

For depth of field, it's a bit more complicated. With a given aperture and focal length, an off-focus scene element projects onto a circle of diffusion (COD) with a fixed size (determined by geometry).

Depth of field is based on the COD that you are willing to accept as "sharp", and here comes the influence of the sensor size, as you'd typically express that as fraction of the image size instead of absolute values. If you accept a 30µm circle as "sharp" with a full-frame sensor, you'd typically reduce that to 20µm with a crop sensor, as you have to do a bigger enlargement from the crop sensor.

While the same scene still gives you the same COD when measured in micro-meters, it gives you a larger one, measured in percent of the image size.

Requiring a smaller COD directly means a smaller DOF, and it scales rather linearly. So, if you start with a DOF formula created for a full-frame sensor, you'd adjust that by the crop factor.

Personally and for clarity, I'd use a DOF formula where I can explicitly insert my COD requirement. To me, this makes the relevant concepts explicit - even with a given sensor size, your COD requirements will vary depending on what you plan to do with the image later, use it for the internet or for a high-class magazine, so I'd stay away from any formula where the COD value isn't used explicitly.

I'd find it rather crude and confusing to express a tighter COD requirement through a modified "effective" aperture.