Every motor has a torque described in kgcm units. 1 kgcm means that the motor can lift 1kg weight on a distance of 1cm from the axis of rotation, or 0.5kg at the distance of 2cm and so on. Using that knowledge and the fact that you are lifting 1kg we can say that arm's length * 1kg must be less than the maximal torque of the motor. The stronger the motor - the longer the arm can be. How strong motor you'll use depends on how much money you'll spend and what length you need for your design.
As for the base of the robot, you need to measure where the center of the mass of the robot will be when the robot is holding the object and make sure that it is somewhere between the two red lines on the image (that is, between the wheels). If the center of mass goes outside of these boundaries, your robot will flip over. If your robot is accelerating, then not only the gravity can flip it but also the force due to acceleration (F=m*a). To check if your robot will flip, add the vectors of gravity force and force due to acceleration, position the beginning of the vector at the center of the mass and check if the vector passes between the wheels. If it doesn't, the robot will flip over.
You can make sure that doesn't happen by keeping your center of mass low (that is, keeping your base heavy), keeping the wheels as separated as possible and not having a too long arm.
I hope this helped you as your question is a bit vague.