No. DOF is not affected by the size of the sensor.
DOF is affected by aperture and magnification. If you stand in the same place, use the same focal length and the same aperture with a full frame camera and a cropped frame camera, your DOF at the sensor plane will be the same. Print the two images to the same size and DOF in the print from the cropped frame sensor will be less.
DOF is affected on cropped frame cameras because in order to get the same scene framed in both cameras you have to either (a) move further away from the subject, (b) use a shorter focal length or (c) a combination of the two. Whatever method you use, magnification is reduced. Lower magnification means greater DOF at the sensor plane. Print the two images to the same size and DOF from the cropped frame sensor will be about the same.
The DOF will be about the same in a print of the same size because the size of the out of focus elements that is considered 'acceptably sharp' gets smaller as the film/sensor format gets smaller on a relative basis.
Example:
Assume: 50mm lens, f8
Distance to subject: 10 feet (full frame), 20 feet (2X crop)
Circle of Confusion size: .025mm (full frame), .013 (2X crop)
DoF (at sensor plane): 5.09ft (full frame), 10.75ft (2X crop)
You have to double the print size of the 2X crop image to make it the same size as the print from the full frame camera. Doubling the print size will double the size of the CoC, making it the same as the full frame camera and halving the DoF in the print of the 2X crop image. You're essentially making magnification of the two images the same again by printing them the same size.